I. OF EUCLID'S "ELEMENTS."
S 6. According to the third postulate it is possible to draw in any plane a circle which has its centre at any given point, and its radius equal to the distance of this point from any other point given in the plane. This makes it possible (Prop. 1) to construct on a given line AB an equilateral triangle, by drawing first a circle with A as centre and AB as radius, and then a circle with B as centre and BA as radius. The point where these circles intersect--that they intersect Euclid quietly assumes--is the vertex of the required triangle. Euclid does not suppose, however, that a circle may be drawn which has its radius equal to the distance between any two points unless one of the points be the centre. This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space. Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set off a distance equal to the length of another straight line given anywhere in the plane. This is done in two steps. It is shown in Prop. 2 how a straight line may be drawn from a given point equal in length to another given straight line not drawn from that point. And then the problem itself is solved in Prop. 3, by drawing first through the given point some straight line of the required length, and then about the same point as centre a circle having this length as radius. This circle will cut off from the given straight line a length equal to the required one. Nowadays, instead of going through this long process, we take a pair of compasses and set off the given length by its aid. This assumes that we may move a length about without changing it. But Euclid has not assumed it, and this proceeding would be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though without stating it.
S 7. We now come (in Prop. 4) to the first theorem. It is the fundamental theorem of Euclid's whole system, there being only a very few propositions (like Props. 13, 14, 15, I.), except those in the 5th book and the first half of the 11th, which do not depend upon it. It is stated very accurately, though somewhat clumsily, as follows:--
If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, namely, those to which the equal sides are opposite.
That is to say, the triangles are "identically" equal, and one may be considered as a copy of the other. The proof is very simple. The first triangle is taken up and placed on the second, so that the parts of the triangles which are known to be equal fall upon each other. It is then easily seen that also the remaining parts of one coincide with those of the other, and that they are therefore equal. This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so. The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.
If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop. 5), if two sides of a triangle are equal, then the angles opposite these sides are equal.
Euclid's proof is somewhat complicated, and a stumbling-block to many schoolboys. The proof becomes much simpler if we consider the isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC, and once as a triangle CAB; and now remember that AB, AC in the first are equal respectively to AC, AB in the second, and the angles included by these sides are equal. Hence the triangles are equal, and the angles in the one are equal to those in the other, viz. those which are opposite equal sides, i.e. angle ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC and AB in the two triangles.
There follows the converse theorem (Prop. 6). If two angles in a triangle are equal, then the sides opposite them are equal,--i.e. the triangle is isosceles. The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem. It assumes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e. to a conclusion which is in contradiction to a proposition proved before--that therefore the assumption made cannot be true, and hence that the theorem is true. It is often stated that Euclid invented this kind of proof, but the method is most likely much older.
S 8. It is next proved that two triangles which have the three sides of the one equal respectively to those of the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides. This is Prop. 8, Prop. 7 containing only a first step towards its proof.
These theorems allow now of the solution of a number of problems, viz.:--
To bisect a given angle (Prop. 9).
To bisect a given finite straight line (Prop. 10).
To draw a straight line perpendicularly to a given straight line through a given point in it (Prop. 11), and also through a given point not in it (Prop. 12).
The solutions all depend upon properties of isosceles triangles.
S 9. The next three theorems relate to angles only, and might have been proved before Prop. 4, or even at the very beginning. The first (Prop. 13) says, The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles. This theorem would have been unnecessary if Euclid had admitted the notion of an angle such that its two limits are in the same straight line, and had besides defined the sum of two angles.
Its converse (Prop. 14) is of great use, inasmuch as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the other. So also is
Prop. 15. If two straight lines cut one another, the vertical or opposite angles shall be equal.
S 10. Euclid returns now to properties of triangles. Of great importance for the next steps (though afterwards superseded by a more complete theorem) is
Prop. 16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.
Prop. 17. Any two angles of a triangle are together less than two right angles, is an immediate consequence of it. By the aid of these two, the following fundamental properties of triangles are easily proved:--
Prop. 18. The greater side of every triangle has the greater angle opposite to it;
Its converse, Prop. 19. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it;
Prop. 20. Any two sides of a triangle are together greater than the third side;
And also Prop. 21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.
S 11. Having solved two problems (Props. 22, 23), he returns to two triangles which have two sides of the one equal respectively to two sides of the other. It is known (Prop. 4) that if the included angles are equal then the third sides are equal; and conversely (Prop. 8), if the third sides are equal, then the angles included by the first sides are equal. From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal. Euclid now completes this knowledge by proving, that "if the included angles are not equal, then the third side in that triangle is the greater which contains the greater angle"; and conversely, that "if the third sides are unequal, that triangle contains the greater angle which contains the greater side." These are Prop. 24 and Prop. 25.
S 12. The next theorem (Prop. 26) says that if two triangles have one side and two angles of the one equal respectively to one side and two angles of the other, viz. in both triangles either the angles adjacent to the equal side, or one angle adjacent and one angle opposite it, then the two triangles are identically equal.
This theorem belongs to a group with Prop. 4 and Prop. 8. Its first case might have been given immediately after Prop. 4, but the second case requires Prop. 16 for its proof.
S 13. We come now to the investigation of parallel straight lines, i.e. of straight lines which lie in the same plane, and cannot be made to meet however far they be produced either way. The investigation which starts from Prop. 16, will become clearer if a few names be explained which are not all used by Euclid. If two straight lines be cut by a third, the latter is now generally called a "transversal" of the figure. It forms at the two points where it cuts the given lines four angles with each. Those of the angles which lie between the given lines are called interior angles, and of these, again, any two which lie on opposite sides of the transversal but one at each of the two points are called "alternate angles."
We may now state Prop. 16 thus:--If two straight lines which meet are cut by a transversal, their alternate angles are unequal. For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it.
From this follows at once the theorem contained in Prop. 27. If two straight lines which are cut by a transversal make alternate angles equal, the lines cannot meet, however far they be produced, hence they are parallel. This proves the existence of parallel lines.
Prop. 28 states the same fact in different forms. If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight lines shall be parallel to one another.
Hence we know that, "if two straight lines which are cut by a transversal meet, their alternate angles are not equal"; and hence that, "if alternate angles are equal, then the lines are parallel."
The question now arises, Are the propositions converse to these true or not? That is to say, "If alternate angles are unequal, do the lines meet?" And "if the lines are parallel, are alternate angles necessarily equal?"
The answer to either of these two questions implies the answer to the other. But it has been found impossible to prove that the negation or the affirmation of either is true.
The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative. This gives his last axiom (12), which we quote in his own words.
Axiom 12.--If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.
The answer to the second of the above questions follows from this, and gives the theorem Prop. 29:--If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior and opposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.
S 14. With this a new part of elementary geometry begins. The earlier propositions are independent of this axiom, and would be true even if a wrong assumption had been made in it. They all relate to figures in a plane. But a plane is only one among an infinite number of conceivable surfaces. We may draw figures on any one of them and study their properties. We may, for instance, take a sphere instead of the plane, and obtain "spherical" in the place of "plane" geometry. If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last all hold, then all propositions up to the 28th will be true for these figures. This is the case in spherical geometry if we substitute "shortest line" or "great circle" for "straight line," "small circle" for "circle," and if, besides, we limit all figures to a part of the sphere which is less than a hemisphere, so that two points on it cannot be opposite ends of a diameter, and therefore determine always one and only one great circle.
For spherical triangles, therefore, all the important propositions 4, 8, 26; 5 and 6; and 18, 19 and 20 will hold good.
This remark will be sufficient to show the impossibility of proving Euclid's last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surface, where the other axioms do hold, whilst parallels do not even exist.
It follows that the axiom in question states an inherent difference between the plane and other surfaces, and that the plane is only fully characterized when this axiom is added to the other assumptions.
S 15. The introduction of the new axiom and of parallel lines leads to a new class of propositions.
After proving (Prop. 30) that "two lines which are each parallel to a third are parallel to each other," we obtain the new properties of triangles contained in Prop. 32. Of these the second part is the most important, viz. the theorem, The three interior angles of every triangle are together equal to two right angles.
As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons, they are given in English editions as corollaries to Prop. 32.
These theorems do not hold for spherical figures. The sum of the interior angles of a spherical triangle is always greater than two right angles, and increases with the area.
S 16. The theory of parallels as such may be said to be finished with Props. 33 and 34, which state properties of the parallelogram, i.e. of a quadrilateral formed by two pairs of parallels. They are--
Prop. 33. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel; and
Prop. 34. The opposite sides and angles of a parallelogram are equal to one another, and the diameter (diagonal) bisects the parallelogram, that is, divides it into two equal parts.
S 17. The rest of the first book relates to areas of figures.
The theory is made to depend upon the theorems--
Prop. 35. Parallelograms on the same base and between the same parallels are equal to one another; and
Prop. 36. Parallelograms on equal bases and between the same parallels are equal to one another.
As each parallelogram is bisected by a diagonal, the last theorems hold also if the word parallelogram be replaced by "triangle," as is done in Props. 37 and 38.
It is to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the same straight line.
The theorems converse to the last form the contents of the next three propositions, viz.: Props, 40 and 41.--Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.
That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.
S 18. To compare areas of other figures, Euclid shows first, in Prop. 42, how to draw a parallelogram which is equal in area to a given triangle, and has one of its angles equal to a given angle. If the given angle is right, then the problem is solved to draw a "rectangle" equal in area to a given triangle.
Next this parallelogram is transformed into another parallelogram, which has one of its sides equal to a given straight line, whilst its angles remain unaltered. This may be done by aid of the theorem in
Prop. 43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.
Thus the problem (Prop. 44) is solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).
As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon. For each of the triangles into which the polygon has been divided, a parallelogram may be constructed, having one side equal to a given straight line and one angle equal to a given angle. If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length. This is done in Prop. 45.
Herewith a means is found to compare areas of different polygons. We need only construct two rectangles equal in area to the given polygons, and having each one side of given length. By comparing the unequal sides we are enabled to judge whether the areas are equal, or which is the greater. Euclid does not state this consequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.
Prop. 46 is: To describe a square on a given straight line.
S 19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times. It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name. If we call that side in a right-angled triangle which is opposite the right angle the hypotenuse, we may state it as follows:--
Theorem of Pythagoras (Prop. 47).--In every right-angled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides.
And conversely--
Prop. 48. If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.
On this theorem (Prop. 47) almost all geometrical measurement depends, which cannot be directly obtained.
Book II
S 20. The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares. Their true significance is best seen by stating them in an algebraic form. This is often done by expressing the lengths of lines by aid of numbers, which tell how many times a chosen unit is contained in the lines. If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being the square on the unit line. If, however, no such unit can be found, this process requires that connexion between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible. But there exists another way of connecting these propositions with algebra, based on modern notions which seem destined greatly to change and to simplify mathematics. We shall introduce here as much of it as is required for our present purpose.
At the beginning of the second book we find a definition according to which "a rectangle is said to be 'contained' by the two sides which contain one of its right angles"; in the text this phraseology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines.
We shall denote a finite straight line by a single small letter, a, b, c, ... x, and the area of the rectangle contained by two lines a and b by ab, and this we shall call the product of the two lines a and b. It will be understood that this definition has nothing to do with the definition of a product of numbers.
We define as follows:--
The sum of two straight lines a and b means a straight line c which may be divided in two parts equal respectively to a and b. This sum is denoted by a + b.
The difference of two lines a and b (in symbols, a-b) means a line c which when added to b gives a; that is,
a - b = c if b + c = a.
The product of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b. For aa, which means the square on the line a, we write a^2.
S 21. The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows:--
Prop. 1. a(b + c + d + ... ) = ab + ac + ad + ...
" 2. ab + ac = a^2 if b + c = a.
" 3. a(a + b) = a^2 + ab.
" 4. (a + b)^2 = a^2 + 2ab + b^2.
" 5. (a + b)(a - b) + b^2 = a^2.
" 6. (a + b)(a - b) + b^2 = a^2.
" 7. a^2 + (a - b)^2 = 2a(a - b) + b^2.
" 8. 4(a + b)a + b^2 = (2a + b)^2.
" 9. (a + b)^2 + (a - b)^2 = 2a^2 + 2b^2.
" 10. (a + b)^2 + (a - b)^2 = 2a^2 + 2b^2.
It will be seen that 5 and 6, and also 9 and 10, are identical. In Euclid's statement they do not look the same, the figures being arranged differently.
If the letters a, b, c, ... denoted numbers, it follows from algebra that each of these formulae is true. But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers. To prove them we have to discover the laws which rule the operations introduced, viz. addition and multiplication of segments. This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.
S 22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups and then add these groups. But this also holds for the sum of segments and for the sum of rectangles, as a little consideration shows. That the sum of rectangles has always a meaning follows from the Props. 43-45 in the first book. These laws about addition are reducible to the two--
a + b = b + a (1),
a + (b + c) = a + b + c (2);
or, when expressed for rectangles,
ab + ed = ed + ab (3),
ab + (cd + ef) = ab + cd + ef (4).
The brackets mean that the terms in the bracket have been added together before they are added to another term. The more general cases for more terms may be deduced from the above.
For the product of two numbers we have the law that it remains unaltered if the factors be interchanged. This also holds for our geometrical product. For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area of the rectangle which has b as base and a as altitude. But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude. This gives
ab = ba (5).
In order further to multiply a sum by a number, we have in algebra the rule:--Multiply each term of the sum, and add the products thus obtained. That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases. In symbols this gives, in the simplest case,
a(b + c) = ab + ac \ > (6). and (b + c)a = ba + ca /
To these laws, which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names. He calls the laws expressed in
(1) and (3) the commutative law for addition;
(5) " " " multiplication;
(2) and (4) the associative laws for addition;
(6) the distributive law.
S 23. Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.
The first is proved geometrically, it being one of the fundamental laws. The next two propositions are only special cases of the first. Of the others we shall prove one, viz. the fourth:--
(a + b)^2 = (a + b)(a + b) = (a + b)a + (a + b)b by (6).
But (a + b)a = aa + ba by (6), = aa + ab by (5);
and (a + b)b = ab + bb by (6).
Therefore (a + b)^2 = aa + ab + (ab + bb) \ = aa + (ab + ab) + bb > by (4). = aa + 2ab + bb /
This gives the theorem in question.
In the same manner every one of the first ten propositions is proved.
It will be seen that the operations performed are exactly the same as if the letters denoted numbers.
Props. 5 and 6 may also be written thus--
(a + b)(a - b) = a^2 - b^2.
Prop. 7, which is an easy consequence of Prop. 4, may be transformed. If we denote by c the line a + b, so that
c = a + b, a = c - b,
we get
c^2 + (c - b)^2 = 2c(c - b) + b^2 = 2c^2 - 2bc + b^2.
Subtracting c^2 from both sides, and writing a for c, we get
(a - b)^2 = a^2 - 2ab + b^2.
In Euclid's Elements this form of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.
S 24. The remaining two theorems (Props. 12 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse. They are important theorems in trigonometry, where it is possible to include them in a single theorem.
S 25. There are in the second book two problems, Props. 11 and 14.
If written in the above symbolic language, the former requires to find a line x such that a(a - x) = x^2. Prop. 11 contains, therefore, the solution of a quadratic equation, which we may write x^2 + ax = a^2. The solution is required later on in the construction of a regular decagon.
More important is the problem in the last proposition (Prop. 14). It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation
x^2 = ab.
In Book I., 42-45, it has been shown how a rectangle may be constructed equal in area to a given figure bounded by straight lines. By aid of the new proposition we may therefore now determine a line such that the square on that line is equal in area to any given rectilinear figure, or we can square any such figure.
As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.
The problem of reducing other areas to squares is frequently met with among Greek mathematicians. We need only mention the problem of squaring the circle (see CIRCLE).
In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base. Their altitudes give then a measure of their areas.
The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop. 43, I. This therefore gives a solution of the equation
ab = ux,
where x denotes the unknown altitude.
Book III
S 26. The third book of the Elements relates exclusively to properties of the circle. A circle and its circumference have been defined in Book I., Def. 15. We restate it here in slightly different words:--
Definition.--The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane. This point is called the "centre" of the circle.
Of the new definitions, of which eleven are given at the beginning of the third book, a few only require special mention. The first, which says that circles with equal radii are equal, is in part a theorem, but easily proved by applying the one circle to the other. Or it may be considered proved by aid of Prop. 24, equal circles not being used till after this theorem.
In the second definition is explained what is meant by a line which "touches" a circle. Such a line is now generally called a tangent to the circle. The introduction of this name allows us to state many of Euclid's propositions in a much shorter form.
For the same reason we shall call a straight line joining two points on the circumference of a circle a "chord."
Definitions 4 and 5 may be replaced with a slight generalization by the following:--
Definition.--By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.
S 27. From the definition of a circle it follows that every circle has a centre. Prop. 1 requires to find it when the circle is given, i.e. when its circumference is drawn.
To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz. the one which bisects the parts of the perpendicular bounded by the circle. In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points. The proof therefore is incomplete. The proof of the first part, however, is exact. By drawing two non-parallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.
S 28. In Prop. 2 it is proved that a chord of a circle lies altogether within the circle.
What we have called the first part of Euclid's solution of Prop. 1 may be stated as a theorem:--
Every straight line which bisects a chord, and is at right angles to it, passes through the centre of the circle.
The converse to this gives Prop. 3, which may be stated thus:--
If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.
An easy consequence of this is the following theorem, which is essentially the same as Prop. 4:--
Two chords of a circle, of which neither passes through the centre, cannot bisect each other.
These last three theorems are fundamental for the theory of the circle. It is to be remarked that Euclid never proves that a straight line cannot have more than two points in common with a circumference.
S 29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz:--
Two circles which have a common centre, and whose circumferences have one point in common, coincide.
Or, more in agreement with Euclid's form:--
Two different circles, whose circumferences have a point in common, cannot have the same centre.
That Euclid treats of two cases is characteristic of Greek mathematics.
The next two propositions (7 and 8) again belong together. They may be combined thus:--
If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre. Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them.
Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.
From the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference, this point must be the centre. This is Prop. 9.
As a consequence of this we get
If the circumferences of the two circles have three points in common they coincide.
For in this case the two circles have a common centre, because from the centre of the one three equal lines can be drawn to points on the circumference of the other. But two circles which have a common centre, and whose circumferences have a point in common, coincide. (Compare above statement of Props. 5 and 6.)
This theorem may also be stated thus:--
Through three points only one circumference may be drawn; or, Three points determine a circle.
Euclid does not give the theorem in this form. He proves, however, that the two circles cannot cut another in more than two points (Prop. 10), and that two circles cannot touch one another in more points than one (Prop. 13).
S 30. Propositions 11 and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres. This gives two propositions, because the circles may touch either internally or externally.
S 31. Propositions 14 and 15 relate to the length of chords. The first says that equal chords are equidistant from the centre, and that chords which are equidistant from the centre are equal;
Whilst Prop. 15 compares unequal chords, viz. Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre; and conversely, the greater chord is nearer to the centre.
S 32. In Prop. 16 the tangent to a circle is for the first time introduced. The proposition is meant to show that the straight line at the end point of the diameter and at right angles to it is a tangent. The proposition itself does not state this. It runs thus:--
Prop. 16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.
Corollary.--The straight line at right angles to a diameter drawn through the end point of it touches the circle.
The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.
Prop. 17 solves the problem through a given point, either in the circumference or without it, to draw a tangent to a given circle.
Closely connected with Prop. 16 are Props. 18 and 19, which state (Prop. 18), that the line joining the centre of a circle to the point of contact of a tangent is perpendicular to the tangent; and conversely (Prop. 19), that the straight line through the point of contact of, and perpendicular to, a tangent to a circle passes through the centre of the circle.
S 33. The rest of the book relates to angles connected with a circle, viz. angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference. Between these two kinds of angles exists the important relation expressed as follows:--
Prop. 20. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.
This is of great importance for its consequences, of which the two following are the principal:--
Prop. 21. The angles in the same segment of a circle are equal to one another;
Prop. 22. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
Further consequences are:--
Prop. 23. On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another;
Prop. 24. Similar segments of circles on equal straight lines are equal to one another.
The problem Prop. 25. A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop. 1, III., in S 27.
S 34. There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs. They are expressed for angles, arcs and chords in equal circles, but they hold also for angles, arcs and chords in the same circle.
The theorems are:--
Prop. 26. In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences;
Prop. 27. (converse to Prop. 26). In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences;
Prop. 28. In equal circles equal straight lines (equal chords) cut off equal arcs, the greater equal to the greater, and the less equal to the less;
Prop. 29 (converse to Prop. 28). In equal circles equal arcs are subtended by equal straight lines.
S 35. Other important consequences of Props. 20-22 are:--
Prop. 31. In a circle the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle;
Prop. 32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.
S 36. Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them:--
Prop. 30. To bisect a given arc, that is, to divide it into two equal parts;
Prop. 33. On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle;
Prop. 34. From a given circle to cut off a segment containing an angle equal to a given rectilineal angle.
S 37. If we draw chords through a point A within a circle, they will each be divided by A into two segments. Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken. The value of this rectangle changes, of course, with the position of A.
A similar theorem holds if the point A be taken without the circle. On every straight line through A, which cuts the circle in two points B and C, we have two segments AB and AC, and the rectangles contained by them are again equal to one another, and equal to the square on a tangent drawn from A to the circle.
The first of these theorems gives Prop. 35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36. The first two theorems may be combined in one:--
If through a point A in the plane of a circle a straight line be drawn cutting the circle in B and C, then the rectangle AB.AC has a constant value so long as the point A be fixed; and if from A a tangent AD can be drawn to the circle, touching at D, then the above rectangle equals the square on AD.
Prop. 37 may be stated thus:--
If from a point A without a circle a line be drawn cutting the circle in B and C, and another line to a point D on the circle, and AB.AC = AD^2, then the line AD touches the circle at D.
It is not difficult to prove also the converse to the general proposition as above stated. This proposition and its converse may be expressed as follows:--
If four points ABCD be taken on the circumference of a circle, and if the lines AB, CD, produced if necessary, meet at E, then
EA.EB = EC.ED;
and conversely, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth.
That a circle may always be drawn through three points, provided that they do not lie in a straight line, is proved only later on in Book IV.
Book IV
S 38. The fourth book contains only problems, all relating to the construction of triangles and polygons inscribed in and circumscribed about circles, and of circles inscribed in or circumscribed about triangles and polygons. They are nearly all given for their own sake, and not for future use in the construction of figures, as are most of those in the former books. In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle. Instead, however, of saying that one figure is described about another, it is now generally said that the one figure is circumscribed about the other. We may then state the definitions 3 or 4 thus:--
Definition.--A polygon is said to be inscribed in a circle, and the circle is said to be circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the circle.
And definitions 5 and 6 thus:--
Definition.--A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the polygon are tangents to the circle.
S 39. The first problem is merely constructive. It requires to draw in a given circle a chord equal to a given straight line, which is not greater than the diameter of the circle. The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the circumference. This may be said of almost all problems in this book, especially of the next two. They are:--
Prop. 2. In a given circle to inscribe a triangle equiangular to a given triangle;
Prop. 3. About a given circle to circumscribe a triangle equiangular to a given triangle.
S 40. Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found.
Prop. 4. To inscribe a circle in a given triangle.
The result is that the problem has always a solution, viz. the centre of the circle is the point where the bisectors of two of the interior angles of the triangle meet. The solution shows, though Euclid does not state this, that the problem has but one solution; and also,
The three bisectors of the interior angles of any triangle meet in a point, and this is the centre of the circle inscribed in the triangle.
The solutions of most of the other problems contain also theorems. Of these we shall state those which are of special interest; Euclid does not state any one of them.
S 41. Prop. 5. To circumscribe a circle about a given triangle.
The one solution which always exists contains the following:--
The three straight lines which bisect the sides of a triangle at right angles meet in a point, and this point is the centre of the circle circumscribed about the triangle.
Euclid adds in a corollary the following property:--
The centre of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is acute-angled, right-angled or obtuse-angled.
S 42. Whilst it is always possible to draw a circle which is inscribed in or circumscribed about a given triangle, this is not the case with quadrilaterals or polygons of more sides. Of those for which this is possible the regular polygons, i.e. polygons which have all their sides and angles equal, are the most interesting. In each of them a circle may be inscribed, and another may be circumscribed about it.
Euclid does not use the word regular, but he describes the polygons in question as equiangular and equilateral. We shall use the name regular polygon. The regular triangle is equilateral, the regular quadrilateral is the square.
Euclid considers the regular polygons of 4, 5, 6 and 15 sides. For each of the first three he solves the problems--(1) to inscribe such a polygon in a given circle; (2) to circumscribe it about a given circle; (3) to inscribe a circle in, and (4) to circumscribe a circle about, such a polygon.
For the regular triangle the problems are not repeated, because more general problems have been solved.
Props. 6, 7, 8 and 9 solve these problems for the square.
The general problem of inscribing in a given circle a regular polygon of n sides depends upon the problem of dividing the circumference of a circle into n equal parts, or what comes to the same thing, of drawing from the centre of the circle n radii such that the angles between consecutive radii are equal, that is, to divide the space about the centre into n equal angles. Thus, if it is required to inscribe a square in a circle, we have to draw four lines from the centre, making the four angles equal. This is done by drawing two diameters at right angles to one another. The ends of these diameters are the vertices of the required square. If, on the other hand, tangents be drawn at these ends, we obtain a square circumscribed about the circle.
S 43. To construct a regular pentagon, we find it convenient first to construct a regular decagon. This requires to divide the space about the centre into ten equal angles. Each will be 1/10th of a right angle, or 1/5th of two right angles. If we suppose the decagon constructed, and if we join the centre to the end of one side, we get an isosceles triangle, where the angle at the centre equals 1/5th of two right angles; hence each of the angles at the base will be 2/5ths of two right angles, as all three angles together equal two right angles. Thus we have to construct an isosceles triangle, having the angle at the vertex equal to half an angle at the base. This is solved in Prop. 10, by aid of the problem in Prop. 11 of the second book. If we make the sides of this triangle equal to the radius of the given circle, then the base will be the side of the regular decagon inscribed in the circle. This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon. (Prop. 11.)
Euclid does not proceed thus. He wants the pentagon before the decagon. This, however, does not change the real nature of his solution, nor does his solution become simpler by not mentioning the decagon.
Once the regular pentagon is inscribed, it is easy to circumscribe another by drawing tangents at the vertices of the inscribed pentagon. This is shown in Prop. 12.
Props. 13 and 14 teach how a circle may be inscribed in or circumscribed about any given regular pentagon.
S 44. The regular hexagon is more easily constructed, as shown in Prop. 15. The result is that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle.
For this polygon the other three problems mentioned are not solved.
S 45. The book closes with Prop. 16. To inscribe a regular quindecagon in a given circle. If we inscribe a regular pentagon and a regular hexagon in the circle, having one vertex in common, then the arc from the common vertex to the next vertex of the pentagon is 1/5th of the circumference, and to the next vertex of the hexagon is 1/6th of the circumference. The difference between these arcs is, therefore, 1/5 - 1/6 = 1/30th of the circumference. The latter may, therefore, be divided into thirty, and hence also in fifteen equal parts, and the regular quindecagon be described.
S 46. We conclude with a few theorems about regular polygons which are not given by Euclid.
The straight lines perpendicular to and bisecting the sides of any regular polygon meet in a point. The straight lines bisecting the angles in the regular polygon meet in the same point. This point is the centre of the circles circumscribed about and inscribed in the regular polygon.
We can bisect any given arc (Prop. 30, III.). Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed. Hence--
If a regular polygon of n sides has been constructed, then a regular polygon of 2n sides, of 4n, of 8n sides, &c., may also be constructed. Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides. It follows that we can construct regular polygons of
3, 6, 12, 24 sides 4, 8, 16, 32 " 5, 10, 20, 40 " 15, 30, 60, 120 "
The construction of any new regular polygon not included in one of these series will give rise to a new series. Till the beginning of the 19th century nothing was added to the knowledge of regular polygons as given by Euclid. Then Gauss, in his celebrated Arithmetic, proved that every regular polygon of 2^n + 1 sides may be constructed if this number 2^n + 1 be prime, and that no others except those with 2^m(2^n + 1) sides can be constructed by elementary methods. This shows that regular polygons of 7, 9, 13 sides cannot thus be constructed, but that a regular polygon of 17 sides is possible; for 17 = 2^4 + 1. The next polygon is one of 257 sides. The construction becomes already rather complicated for 17 sides.
Book V
S 47. The fifth book of the Elements is not exclusively geometrical. It contains the theory of ratios and proportion of quantities in general. The treatment, as here given, is admirable, and in every respect superior to the algebraical method by which Euclid's theory is now generally replaced. We shall treat the subject in order to show why the usual algebraical treatment of proportion is not really sound. We begin by quoting those definitions at the beginning of Book V. which are most important. These definitions have given rise to much discussion.
The only definitions which are essential for the fifth book are Defs. 1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than useless, and probably not Euclid's, but additions of later editors, of whom Theon of Alexandria was the most prominent. Defs. 10 and 11 belong rather to the sixth book, whilst all the others are merely nominal. The really important ones are 4, 5, 6 and 7.
S 48. To define a magnitude is not attempted by Euclid. The first two definitions state what is meant by a "part," that is, a submultiple or measure, and by a "multiple" of a given magnitude. The meaning of Def. 4 is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude, that is, if they are of the same kind.
Def. 3, which is probably due to Theon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Defs. 5 and 7.
In Def. 5 it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes, and in Def. 7 what we have to understand by a greater or a less ratio. The 6th definition is only nominal, explaining the meaning of the word proportional.
Euclid represents magnitudes by lines, and often denotes them either by single letters or, like lines, by two letters. We shall use only single letters for the purpose. If a and b denote two magnitudes of the same kind, their ratio will be denoted by a : b; if c and d are two other magnitudes of the same kind, but possibly of a different kind from a and b, then if c and d have the same ratio to one another as a and b, this will be expressed by writing--
a : b :: c : d.
Further, if m is a (whole) number, ma shall denote the multiple of a which is obtained by taking it m times.
S 49. The whole theory of ratios is based on Def. 5.
Def. 5. The first of four magnitudes is said to have the same ratio to the second that the third has to the fourth when, any equimultiples whatever of the first and the third being taken, and any equimultiples whatever of the second and the fourth, if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; and if the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth; and if the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth.
It will be well to show at once in an example how this definition can be used, by proving the first part of the first proposition in the sixth book. Triangles of the same altitude are to one another as their bases, or if a and b are the bases, and [alpha] and [beta] the areas, of two triangles which have the same altitude, then a : b :: [alpha] : [beta].
To prove this, we have, according to Definition 5, to show--
if ma > nb, then m[alpha] > n[beta], if ma = nb, then m[alpha] = n[beta], if ma < nb, then m[alpha] < n[beta].
That this is true is in our case easily seen. We may suppose that the triangles have a common vertex, and their bases in the same line. We set off the base a along the line containing the bases m times; we then join the different parts of division to the vertex, and get m triangles all equal to [alpha]. The triangle on ma as base equals, therefore, m[alpha]. If we proceed in the same manner with the base b, setting it off n times, we find that the area of the triangle on the base nb equals n[beta], the vertex of all triangles being the same. But if two triangles have the same altitude, then their areas are equal if the bases are equal; hence m[alpha] = n[beta] if ma = nb, and if their bases are unequal, then that has the greater area which is on the greater base; in other words, m[alpha] is greater than, equal to, or less than n[beta], according as ma is greater than, equal to, or less than nb, which was to be proved.
S 50. It will be seen that even in this example it does not become evident what a ratio really is. It is still an open question whether ratios are magnitudes which we can compare. We do not know whether the ratio of two lines is a magnitude of the same kind as the ratio of two areas. Though we might say that Def. 5 defines _equal _ratios, still we do not know whether they are equal in the sense of the axiom, that two things which are equal to a third are equal to one another. That this is the case requires a proof, and until this proof is given we shall use the :: instead of the sign = , which, however, we shall afterwards introduce.
As soon as it has been established that all ratios are like magnitudes, it becomes easy to show that, in some cases at least, they are numbers. This step was never made by Greek mathematicians. They distinguished always most carefully between continuous magnitudes and the discrete series of numbers. In modern times it has become the custom to ignore this difference.
If, in determining the ratio of two lines, a common measure can be found, which is contained m times in the first, and n times in the second, then the ratio of the two lines equals the ratio of the two numbers m : n. This is shown by Euclid in Prop. 5, X. But the ratio of two numbers is, as a rule, a fraction, and the Greeks did not, as we do, consider fractions as numbers. Far less had they any notion of introducing irrational numbers, which are neither whole nor fractional, as we are obliged to do if we wish to say that all ratios are numbers. The incommensurable numbers which are thus introduced as ratios of incommensurable quantities are nowadays as familiar to us as fractions; but a proof is generally omitted that we may apply to them the rules which have been established for rational numbers only. Euclid's treatment of ratios avoids this difficulty. His definitions hold for commensurable as well as for incommensurable quantities. Even the notion of incommensurable quantities is avoided in Book V. But he proves that the more elementary rules of algebra hold for ratios. We shall state all his propositions in that algebraical form to which we are now accustomed. This may, of course, be done without changing the character of Euclid's method.
S. 51. Using the notation explained above we express the first propositions as follows:--
Prop. 1. If a = ma', b = mb', c = mc', then a + b + c = m(a' + b' + c').
Prop. 2. If a = mb, and c = md, e = nb, and f = nd,
then a + e is the same multiple of b as c + f is of d, viz.:--
a + e = (m + n)b, and c + f = (m + n)d.
Prop. 3. If a = mb, c = md, then is na the same multiple of b that nc is of d, viz. na = nmb, nc = nmd.
Prop. 4. If a : b :: c : d, then ma : nb :: mc : nd.
Prop. 5. If a = mb, and c = md, then a - c = m(b - d).
Prop. 6. If a = mb, c = md,
then are a - nb and c - nd either equal to, or equimultiples of, b and d, viz. a - nb = (m - n)b and c - nd = (m - n)d, where m - n may be unity.
All these propositions relate to equimultiples. Now follow propositions about ratios which are compared as to their magnitude.
S 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.
The proof is simply this. As a = b we know that ma = mb; therefore
if ma > nc, then mb > nc, if ma = nc, then mb = nc, if ma < nc, then mb < nc,
therefore the first proportion holds by Definition 5.
Prop. 8. If a > b, then a : c > b : c, and c : a < c : b.
The proof depends on Definition 7.
Prop. 9 (converse to Prop. 7). If a : c :: b : c, or if c : a :: c : b, then a = b.
Prop. 10 (converse to Prop. 8). If a : c > b : c, then a > b, and if c : a < c : b, then a < b.
Prop. 11. If a : b :: c : d, and a : b :: e : f, then c : d :: e : f.
In words, if too ratios are equal to a third, they are equal to one another. After these propositions have been proved, we have a right to consider a ratio as a magnitude, for only now can we consider a ratio as something for which the axiom about magnitudes holds: things which are equal to a third are equal to one another.
We shall indicate this by writing in future the sign = instead of ::. The remaining propositions, which explain themselves, may then be stated as follows:
S 53. Prop. 12. If a : b = c : d = e : f, then a + c + e : b + d + f = a : b.
Prop. 13. If a : b = c : d and c : d > e : f, then a : b > e : f.
Prop. 14. If a : b = c : d, and a > c, then b > d.
Prop. 15. Magnitudes have the same ratio to one another that their equimultiples have--
ma : mb = a : b.
Prop. 16. If a, b, c, d are magnitudes of the same kind, and if a : b = c : d, then a : c = b : d.
Prop. 17. If a + b : b = c + d : d, then a : b = c : d.
Prop. 18 (converse to 17). If a : b = c : d then a + b : b = c + d : d.
Prop. 19. If a, b, c, d are quantities of the same kind, and if a : b = c : d, then a - c : b - d = a : b.
S 54. Prop. 20. If there be three magnitudes, and another three, which have the same ratio, taken two and two, then if the first be greater than the third, the fourth shall be greater than the sixth: and if equal, equal; and if less, less.
If we understand by
a : b : c : d : e : ... = a' : b' : c' : d' : e' : ...
that the ratio of any two consecutive magnitudes on the first side equals that of the corresponding magnitudes on the second side, we may write this theorem in symbols, thus:--
If a, b, c be quantities of one, and d, e, f magnitudes of the same or any other kind, such that
a : b : c = d : e : f, and if a > c, then d > f, but if a = c, then d = f, and if a < c, then d < f.
Prop. 21. If a : b = e : f and b : c = d : e, or if a : b : c = 1/f : 1/e : 1/d, and if a > c, then d > f, but if a = c, then d = f, and if a < c, then d < f.
By aid of these two propositions the following two are proved.
S 55. Prop. 22. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.
We may state it more generally, thus:
If a : b : c : d : e: ... = a' : b' : c' : d' : e' : ... ,
then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For instance--
a : c = a' : c'; b : e = b' : e', &c.
Prop. 23 we state only in symbols, viz.:--
If a : b : c : d : e : ... = 1/a' : 1/b' : 1/c' : 1/d' : 1/e' ...,
then a : c = c' : a', b : e = e' : b',
and so on.
Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then
a + e : b = c + f : d.
Some of the proportions which are considered in the above propositions have special names. These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.
S 56. The last proposition in the fifth book is of a different character.
Prop. 25. If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together. In symbols--
If a, b, c, d be magnitudes of the same kind, and if a : b = c : d, and if a is the greatest, hence d the least, then a + d > b + c.
S 57. We return once again to the question. What is a ratio? We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude. It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, How often is one line contained in another? But the answer to this question is given by a number, at least in some cases, and in all cases if we admit incommensurable numbers. Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers. In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.
Book VI
S 58. The sixth book contains the theory of similar figures. After a few definitions explaining terms, the first proposition gives the first application of the theory of proportion.
Prop. 1. Triangles and parallelograms of the same altitude are to one another as their bases.
The proof has already been considered in S 49.
From this follows easily the important theorem
Prop. 2. If a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides, or those sides produced, proportionally; and if the sides or the sides produced be cut proportionally, the straight line which joins the points of section shall be parallel to the remaining side of the triangle.
S 59. The next proposition, together with one added by Simson as Prop. A, may be expressed more conveniently if we introduce a modern phraseology, viz. if in a line AB we assume a point C between A and B, we shall say that C divides AB internally in the ratio AC : CB; but if C be taken in the line AB produced, we shall say that AB is divided externally in the ratio AC : CB.
The two propositions then come to this:
Prop. 3. The bisector of an angle in a triangle divides the opposite side internally in a ratio equal to the ratio of the two sides including that angle; and conversely, if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex.
Simson's Prop. A. The line which bisects an exterior angle of a triangle divides the opposite side externally in the ratio of the other sides; and conversely, if a line through the vertex of a triangle divide the base externally in the ratio of the sides, then it bisects an exterior angle at the vertex of the triangle.
If we combine both we have--
The two lines which bisect the interior and exterior angles at one vertex of a triangle divide the opposite side internally and externally in the same ratio, viz. in the ratio of the other two sides.
S 60. The next four propositions contain the theory of similar triangles, of which four cases are considered. They may be stated together.
Two triangles are similar,--
1. (Prop. 4). If the triangles are equiangular:
2. (Prop. 5). If the sides of the one are proportional to those of the other;
3. (Prop. 6). If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal;
4. (Prop. 7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right or both obtuse; homologous sides being in each case those which are opposite equal angles.
An important application of these theorems is at once made to a right-angled triangle, viz.:--
Prop. 8. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Corollary.--From this it is manifest that the perpendicular drawn from the right angle of a right-angled triangle to the base is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.
S 61. There follow four propositions containing problems, in language slightly different from Euclid's, viz.:--
Prop. 9. To divide a straight line into a given number of equal parts.
Prop. 10. To divide a straight line in a given ratio.
Prop. 11. To find a third proportional to two given straight lines.
Prop. 12. To find a fourth proportional to three given straight lines.
Prop. 13. To find a mean proportional between two given straight lines.
The last three may be written as equations with one unknown quantity--viz. if we call the given straight lines a, b, c, and the required line x, we have to find a line x so that
Prop. 11. a : b = b : x;
Prop. 12. a : b = c : x;
Prop. 13. a : x = x : b.
We shall see presently how these may be written without the signs of ratios.
S 62. Euclid considers next proportions connected with parallelograms and triangles which are equal in area.
Prop. 14. Equal parallelograms which have one angle of the one equal to one angle of the other have their sides about the equal angles reciprocally proportional; and parallelograms which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
Prop. 15. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.
The latter proposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the figure belonging to the latter.
It is worth noticing that the lines FE and DG are parallel. We may state therefore the theorem--
If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.
S 63. A most important theorem is
Prop. 16. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals.
In symbols, if a, b, c, d are the four lines, and if a : b = c : d, then ad = bc; and conversely, if ad = bc, then a : b = c : d,
where ad and bc denote (as in S 20), the areas of the rectangles contained by a and d and by b and c respectively.
This allows us to transform every proportion between four lines into an equation between two products.
It shows further that the operation of forming a product of two lines, and the operation of forming their ratio are each the inverse of the other.
If we now define a quotient a/b of two lines as the number which multiplied into b gives a, so that
a -- b = a, b
we see that from the equality of two quotients
a c -- = -- b d
follows, if we multiply both sides by bd,
a c -- b.d = -- d.b, b d
ad = cb.
But from this it follows, according to the last theorem, that
a : b = c : d.
Hence we conclude that the quotient a/b and the ratio a : b are different forms of the same magnitude, only with this important difference that the quotient a/b would have a meaning only if a and b have a common measure, until we introduce incommensurable numbers, while the ratio a : b has always a meaning, and thus gives rise to the introduction of incommensurable numbers.
Thus it is really the theory of ratios in the fifth book which enables us to extend the geometrical calculus given before in connexion with Book II. It will also be seen that if we write the ratios in Book V. as quotients, or rather as fractions, then most of the theorems state properties of quotients or of fractions.
S 64. Prop. 17. If three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean; and conversely, is only a special case of 16. After the problem, Prop. 18, On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure, there follows another fundamental theorem:
Prop. 19. Similar triangles are to one another in the duplicate ratio of their homologous sides. In other words, the areas of similar triangles are to one another as the squares on homologous sides. This is generalized in:
Prop. 20. Similar polygons may be divided into the same number of similar triangles, having the same ratio to one another that the polygons have; and the polygons are to one another in the duplicate ratio of their homologous sides.
S 65. Prop. 21. Rectilineal figures which are similar to the same rectilineal figure are also similar to each other, is an immediate consequence of the definition of similar figures. As similar figures may be said to be equal in "shape" but not in "size," we may state it also thus:
"Figures which are equal in shape to a third are equal in shape to each other."
Prop. 22. If four straight lines be proportionals, the similar rectilineal figures similarly described on them shall also be proportionals; and if the similar rectilineal figures similarly described on four straight lines be proportionals, those straight lines shall be proportionals.
This is essentially the same as the following:--
If a : b = c : d, then a^2 : b^2 = c^2 : d^2.
S 66. Now follows a proposition which has been much discussed with regard to Euclid's exact meaning in saying that a ratio is compounded of two other ratios, viz.:
Prop. 23. Parallelograms which are equiangular to one another, have to one another the ratio which is compounded of the ratios of their sides.
The proof of the proposition makes its meaning clear. In symbols the ratio a : c is compounded of the two ratios a : b and b : c, and if a : b = a' : b', b : c = b" : c", then a : c is compounded of a' : b' and b" : c".
If we consider the ratios as numbers, we may say that the one ratio is the product of those of which it is compounded, or in symbols,
a a b a' b" a a' b b" -- = -- . -- = -- . --, if -- = -- and -- = --. c b c b' c" b b' c c"
The theorem in Prop. 23 is the foundation of all mensuration of areas. From it we see at once that two rectangles have the ratio of their areas compounded of the ratios of their sides.
If A is the area of a rectangle contained by a and b, and B that of a rectangle contained by c and d, so that A = ab, B = cd, then A : B = ab : cd, and this is, the theorem says, compounded of the ratios a : c and b : d. In forms of quotients,
a b ab -- . -- = --. c d cd
This shows how to multiply quotients in our geometrical calculus.
Further, Two triangles have the ratios of their areas compounded of the ratios of their bases and their altitude. For a triangle is equal in area to half a parallelogram which has the same base and the same altitude.
S 67. To bring these theorems to the form in which they are usually given, we assume a straight line u as our unit of length (generally an inch, a foot, a mile, &c.), and determine the number [alpha] which expresses how often u is contained in a line a, so that [alpha] denotes the ratio a : u whether commensurable or not, and that a = [alpha]u. We call this number [alpha] the numerical value of a. If in the same manner [beta] be the numerical value of a line b we have
a : b = [alpha] : [beta];
in words: The ratio of two lines (and of two like quantities in general) is equal to that of their numerical values.
This is easily proved by observing that a = [alpha]u, b = [beta]u, therefore a : b = [alpha]u : [beta]u, and this may without difficulty be shown to equal [alpha] : [beta].
If now a, b be base and altitude of one, a', b' those of another parallelogram, [alpha], [beta] and [alpha]', [beta]' their numerical values respectively, and A, A' their areas, then
A a b [alpha] [beta] [alpha][beta] -- = -- . -- = -------- . ------ = ---------------. A' a' b' [alpha]' [beta]' [alpha]'[beta]'
In words: The areas of two parallelograms are to each other as the products of the numerical values of their bases and altitudes.
If especially the second parallelogram is the unit square, i.e. a square on the unit of length, then [alpha]' = [beta]' = 1, A' = u^2, and we have
A -- = [alpha][beta] or A = [alpha][beta] . u^2. A'
This gives the theorem: The number of unit squares contained in a parallelogram equals the product of the numerical values of base and altitude, and similarly the number of unit squares contained in a triangle equals half the product of the numerical values of base and altitude.
This is often stated by saying that the area of a parallelogram is equal to the product of the base and the altitude, meaning by this product the product of the numerical values, and not the product as defined above in S 20.
S 68. Propositions 24 and 26 relate to parallelograms about diagonals, such as are considered in Book I., 43. They are--
Prop. 24. Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another; and its converse (Prop. 26), If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter.
Between these is inserted a problem.
Prop. 25. To describe a rectilineal figure which shall be similar to one given rectilinear figure, and equal to another given rectilineal figure.
S 69. Prop. 27 contains a theorem relating to the theory of maxima and minima. We may state it thus:
Prop. 27. If a parallelogram be divided into two by a straight line cutting the base, and if on half the base another parallelogram be constructed similar to one of those parts, then this third parallelogram is greater than the other part.
Of far greater interest than this general theorem is a special case of it, where the parallelograms are changed into rectangles, and where one of the parts into which the parallelogram is divided is made a square; for then the theorem changes into one which is easily recognized to be identical with the following:--
Of all rectangles which have the same perimeter the square has the greatest area.
This may also be stated thus:--
Of all rectangles which have the same area the square has the least perimeter.
S 70. The next three propositions contain problems which may be said to be solutions of quadratic equations. The first two are, like the last, involved in somewhat obscure language. We transcribe them as follows:
Problem.--To describe on a given base a parallelogram, and to divide it either internally (Prop. 28) or externally (Prop. 29) from a point on the base into two parallelograms, of which the one has a given size (is equal in area to a given figure), whilst the other has a given shape (is similar to a given parallelogram).
If we express this again in symbols, calling the given base a, the one part x, and the altitude y, we have to determine x and y in the first case from the equations
(a - x)y = k^2,
x p -- = --, y q
k^2 being the given size of the first, and p and q the base and altitude of the parallelogram which determine the shape of the second of the required parallelograms.
If we substitute the value of y, we get
pk^2 (a - x)x = ----, q
or,
ax - x^2 = b^2,
where a and b are known quantities, taking b^2 = pk^2/q.
The second case (Prop. 29) gives rise, in the same manner, to the quadratic
ax + x^2 = b^2.
The next problem--
Prop. 30. To cut a given straight line in extreme and mean ratio, leads to the equation
ax + x^2 = a^2.
This is, therefore, only a special case of the last, and is, besides, an old acquaintance, being essentially the same problem as that proposed in II. 11.
Prop. 30 may therefore be solved in two ways, either by aid of Prop. 29 or by aid of II. 11. Euclid gives both solutions.
S 71. Prop. 31 (Theorem). In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly-described figures on the sides containing the right angle,--is a pretty generalization of the theorem of Pythagoras (I. 47).
Leaving out the next proposition, which is of little interest, we come to the last in this book.
Prop. 33. In equal circles angles, whether at the centres or the circumferences, have the same ratio which the arcs on which they stand have to one another; so also have the sectors.
Of this, the part relating to angles at the centre is of special importance; it enables us to measure angles by arcs.
With this closes that part of the Elements which is devoted to the study of figures in a plane.
Book XI
S 72. In this book figures are considered which are not confined to a plane, viz. first relations between lines and planes in space, and afterwards properties of solids.
Of new definitions we mention those which relate to the perpendicularity and the inclination of lines and planes.
Def. 3. A straight line is perpendicular, or at right angles, to a plane when it makes right angles with every straight line meeting it in that plane.
The definition of perpendicular planes (Def. 4) offers no difficulty. Euclid defines the inclination of lines to planes and of planes to planes (Defs. 5 and 6) by aid of plane angles, included by straight lines, with which we have been made familiar in the first books.
The other important definitions are those of parallel planes, which never meet (Def. 8), and of solid angles formed by three or more planes meeting in a point (Def. 9).
To these we add the definition of a line parallel to a plane as a line which does not meet the plane.
S 73. Before we investigate the contents of Book XI., it will be well to recapitulate shortly what we know of planes and lines from the definitions and axioms of the first book. There a plane has been defined as a surface which has the property that every straight line which joins two points in it lies altogether in it. This is equivalent to saying that a straight line which has two points in a plane has all points in the plane. Hence, a straight line which does not lie in the plane cannot have more than one point in common with the plane. This is virtually the same as Euclid's Prop. 1, viz.:--
Prop. 1. One part of a straight line cannot be in a plane and another part without it.
It also follows, as was pointed out in S 3, in discussing the definitions of Book I., that a plane is determined already by one straight line and a point without it, viz. if all lines be drawn through the point, and cutting the line, they will form a plane.
This may be stated thus:--
A plane is determined--
1st, By a straight line and a point which does not lie on it;
2nd, By three points which do not lie in a straight line; for if two of these points be joined by a straight line we have case 1;
3rd, By two intersecting straight lines; for the point of intersection and two other points, one in each line, give case 2;
4th, By two parallel lines (Def. 35, I.).
The third case of this theorem is Euclid's
Prop. 2. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.
And the fourth is Euclid's
Prop. 7. If two straight lines be parallel, the straight line drawn from any point in one to any point in the other is in the same plane with the parallels. From the definition of a plane further follows
Prop. 3. If two planes cut one another, their common section is a straight line.
S 74. Whilst these propositions are virtually contained in the definition of a plane, the next gives us a new and fundamental property of space, showing at the same time that it is possible to have a straight line perpendicular to a plane, according to Def. 3. It states--
Prop. 4. If a straight line is perpendicular to two straight lines in a plane which it meets, then it is perpendicular to all lines in the plane which it meets, and hence it is perpendicular to the plane.
Def. 3 may be stated thus: If a straight line is perpendicular to a plane, then it is perpendicular to every line in the plane which it meets. The converse to this would be
All straight lines which meet a given straight line in the same point, and are perpendicular to it, lie in a plane which is perpendicular to that line.
This Euclid states thus:
Prop. 5. If three straight lines meet all at one point, and a straight line stands at right angles to each of them at that point, the three straight lines shall be in one and the same plane.
S 75. There follow theorems relating to the theory of parallel lines in space, viz.:--
Prop. 6. Any two lines which are perpendicular to the same plane are parallel to each other; and conversely
Prop. 8. If of two parallel straight lines one is perpendicular to a plane, the other is so also.
Prop. 7. If two straight lines are parallel, the straight line which joins any point in one to any point in the other is in the same plane as the parallels. (See above, S 73.)
Prop. 9. Two straight lines which are each of them parallel to the same straight line, and not in the same plane with it, are parallel to one another; where the words, "and not in the same plane with it," may be omitted, for they exclude the case of three parallels in a plane, which has been proved before; and
Prop. 10. If two angles in different planes have the two limits of the one parallel to those of the other, then the angles are equal. That their planes are parallel is shown later on in Prop. 15.
This theorem is not necessarily true, for the angles in question may be supplementary; but then the one angle will be equal to that which is adjacent and supplementary to the other, and this latter angle will also have its limits parallel to those of the first.
From this theorem it follows that if we take any two straight lines in space which do not meet, and if we draw through any point P in space two lines parallel to them, then the angle included by these lines will always be the same, whatever the position of the point P may be. This angle has in modern times been called the angle between the given lines:--
By the angles between two not intersecting lines we understand the angles which two intersecting lines include that are parallel respectively to the two given lines.
S 76. It is now possible to solve the following two problems:--
To draw a straight line perpendicular to a given plane from a given point which lies
1. Not in the plane (Prop. 11).
2. In the plane (Prop. 12).
The second case is easily reduced to the first--viz. if by aid of the first we have drawn any perpendicular to the plane from some point without it, we need only draw through the given point in the plane a line parallel to it, in order to have the required perpendicular given. The solution of the first part is of interest in itself. It depends upon a construction which may be expressed as a theorem.
If from a point A without a plane a perpendicular AB be drawn to the plane, and if from the foot B of this perpendicular another perpendicular BC be drawn to any straight line in the plane, then the straight line joining A to the foot C of this second perpendicular will also be perpendicular to the line in the plane.
The theory of perpendiculars to a plane is concluded by the theorem--
Prop. 13. Through any point in space, whether in or without a plane, only one straight line can be drawn perpendicular to the plane.
S 77. The next four propositions treat of parallel planes. It is shown that planes which have a common perpendicular are parallel (Prop. 14); that two planes are parallel if two intersecting straight lines in the one are parallel respectively to two straight lines in the other plane (Prop. 15); that parallel planes are cut by any plane in parallel straight lines (Prop. 16); and lastly, that any two straight lines are cut proportionally by a series of parallel planes (Prop. 17).
This theory is made more complete by adding the following theorems, which are easy deductions from the last: Two parallel planes have common perpendiculars (converse to 14); and Two planes which are parallel to a third plane are parallel to each other.
It will be noted that Prop. 15 at once allows of the solution of the problem: "Through a given point to draw a plane parallel to a given plane." And it is also easily proved that this problem allows always of one, and only of one, solution.
S 78. We come now to planes which are perpendicular to one another. Two theorems relate to them.
Prop. 18. If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane.
Prop. 19. If two planes which cut one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane.
S 79. If three planes pass through a common point, and if they bound each other, a solid angle of three faces, or a trihedral angle, is formed, and similarly by more planes a solid angle of more faces, or a polyhedral angle. These have many properties which are quite analogous to those of triangles and polygons in a plane. Euclid states some, viz.:--
Prop. 20. If a solid angle be contained by three plane angles, any two of them are together greater than the third.
But the next--
Prop. 21. Every solid angle is contained by plane angles, which are together less than four right angles--has no analogous theorem in the plane.
We may mention, however, that the theorems about triangles contained in the propositions of Book I., which do not depend upon the theory of parallels (that is all up to Prop. 27), have their corresponding theorems about trihedral angles. The latter are formed, if for "side of a triangle" we write "plane angle" or "face" of trihedral angle, and for "angle of triangle" we substitute "angle between two faces" where the planes containing the solid angle are called its faces. We get, for instance, from I. 4, the theorem, If two trihedral angles have the angles of two faces in the one equal to the angles of two faces in the other, and have likewise the angles included by these faces equal, then the angles in the remaining faces are equal, and the angles between the other faces are equal each to each, viz. those which are opposite equal faces. The solid angles themselves are not necessarily equal, for they may be only symmetrical like the right hand and the left.
The connexion indicated between triangles and trihedral angles will also be recognized in
Prop. 22. If every two of three plane angles be greater than the third, and if the straight lines which contain them be all equal, a triangle may be made of the straight lines that join the extremities of those equal straight lines.
And Prop. 23 solves the problem, To construct a trihedral angle having the angles of its faces equal to three given plane angles, any two of them being greater than the third. It is, of course, analogous to the problem of constructing a triangle having its sides of given length.
Two other theorems of this kind are added by Simson in his edition of Euclid's Elements.
S 80. These are the principal properties of lines and planes in space, but before we go on to their applications it will be well to define the word distance. In geometry distance means always "shortest distance"; viz. the distance of a point from a straight line, or from a plane, is the length of the perpendicular from the point to the line or plane. The distance between two non-intersecting lines is the length of their common perpendicular, there being but one. The distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes.
S 81. Parallelepipeds.--The rest of the book is devoted to the study of the parallelepiped. In Prop. 24 the possibility of such a solid is proved, viz.:--
Prop. 24. If a solid be contained by six planes two and two of which are parallel, the opposite planes are similar and equal parallelograms.
Euclid calls this solid henceforth a parallelepiped, though he never defines the word. Either face of it may be taken as base, and its distance from the opposite face as altitude.
Prop. 25. If a solid parallelepiped be cut by a plane parallel to two of its opposite planes, it divides the whole into two solids, the base of one of which shall be to the base of the other as the one solid is to the other.
This theorem corresponds to the theorem (VI. 1) that parallelograms between the same parallels are to one another as their bases. A similar analogy is to be observed among a number of the remaining propositions.
S 82. After solving a few problems we come to
Prop. 28. If a solid parallelepiped be cut by a plane passing through the diagonals of two of the opposite planes, it shall be cut in two equal parts.
In the proof of this, as of several other propositions, Euclid neglects the difference between solids which are symmetrical like the right hand and the left.
Prop. 31. Solid parallelepipeds, which are upon equal bases, and of the same altitude, are equal to one another.
Props. 29 and 30 contain special cases of this theorem leading up to the proof of the general theorem.
As consequences of this fundamental theorem we get
Prop. 32. Solid parallelepipeds, which have the same altitude, are to one another as their bases; and
Prop. 33. Similar solid parallelepipeds are to one another in the triplicate ratio of their homologous sides.
If we consider, as in S 67, the ratios of lines as numbers, we may also say--
The ratio of the volumes of similar parallelepipeds is equal to the ratio of the third powers of homologous sides.
Parallelepipeds which are not similar but equal are compared by aid of the theorem
Prop. 34. The bases and altitudes of equal solid parallelepipeds are reciprocally proportional; and if the bases and altitudes be reciprocally proportional, the solid parallelepipeds are equal.
S 83. Of the following propositions the 37th and 40th are of special interest.
Prop. 37. If four straight lines be proportionals, the similar solid parallelepipeds, similarly described from them, shall also be proportionals; and if the similar parallelepipeds similarly described from four straight lines be proportionals, the straight lines shall be proportionals.
In symbols it says--
If a : b = c : d, then a^3 : b^3 = c^3 : d^3.
Prop. 40 teaches how to compare the volumes of triangular prisms with those of parallelepipeds, by proving that a triangular prism is equal in volume to a parallelepiped, which has its altitude and its base equal to the altitude and the base of the triangular prism.
S 84. From these propositions follow all results relating to the mensuration of volumes. We shall state these as we did in the case of areas. The starting-point is the "rectangular" parallelepiped, which has every edge perpendicular to the planes it meets, and which takes the place of the rectangle in the plane. If this has all its edges equal we obtain the "cube."
If we take a certain line u as unit length, then the square on u is the unit of area, and the cube on u the unit of volume, that is to say, if we wish to measure a volume we have to determine how many unit cubes it contains.
A rectangular parallelepiped has, as a rule, the three edges unequal, which meet at a point. Every other edge is equal to one of them. If a, b, c be the three edges meeting at a point, then we may take the rectangle contained by two of them, say by b and c, as base and the third as altitude. Let V be its volume, V' that of another rectangular parallelepiped which has the edges a', b, c, hence the same base as the first. It follows then easily, from Prop. 25 or 32, that V : V' = a : a'; or in words,
Rectangular parallelepipeds on equal bases are proportional to their altitudes.
If we have two rectangular parallelepipeds, of which the first has the volume V and the edges a, b, c, and the second, the volume V' and the edges a', b', c', we may compare them by aid of two new ones which have respectively the edges a', b, c and a', b', c, and the volumes V1 and V2. We then have
V : V1 = a : a'; V1 : V2 = b : b', V2 : V' = c : c'.
Compounding these, we have
V : V' = (a : a')(b : b')(c : c'),
or
V a b c -- = -- . -- . --. V' a' b' c'
Hence, as a special case, making V' equal to the unit cube U on u we get
V a b c -- = -- . -- . -- = [alpha].[beta].[gamma], U u u u
where [alpha], [beta], [gamma] are the numerical values of a, b, c; that is, The number of unit cubes in a rectangular parallelepiped is equal to the product of the numerical values of its three edges. This is generally expressed by saying the volume of a rectangular parallelepiped is measured by the product of its sides, or by the product of its base into its altitude, which in this case is the same.
Prop. 31 allows us to extend this to any parallelepipeds, and Props. 28 or 40, to triangular prisms.
The volume of any parallelepiped, or of any triangular prism, is measured by the product of base and altitude.
The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds for any prism whatever.
Book XII
S 85. In the last part of Book XI. we have learnt how to compare the volumes of parallelepipeds and of prisms. In order to determine the volume of any solid bounded by plane faces we must determine the volume of pyramids, for every such solid may be decomposed into a number of pyramids.
As every pyramid may again be decomposed into triangular pyramids, it becomes only necessary to determine their volume. This is done by the
Theorem.--Every triangular pyramid is equal in volume to one third of a triangular prism having the same base and the same altitude as the pyramid.
This is an immediate consequence of Euclid's
Prop. 7. Every prism having a triangular base may be divided into three pyramids that have triangular bases, and are equal to one another.
The proof of this theorem is difficult, because the three triangular pyramids into which the prism is divided are by no means equal in shape, and cannot be made to coincide. It has first to be proved that two triangular pyramids have equal volumes, if they have equal bases and equal altitudes. This Euclid does in the following manner. He first shows (Prop. 3) that a triangular pyramid may be divided into four parts, of which two are equal triangular pyramids similar to the whole pyramid, whilst the other two are equal triangular prisms, and further, that these two prisms together are greater than the two pyramids, hence more than half the given pyramid. He next shows (Prop. 4) that if two triangular pyramids are given, having equal bases and equal altitudes, and if each be divided as above, then the two triangular prisms in the one are equal to those in the other, and each of the remaining pyramids in the one has its base and altitude equal to the base and altitude of the remaining pyramids in the other. Hence to these pyramids the same process is again applicable. We are thus enabled to cut out of the two given pyramids equal parts, each greater than half the original pyramid. Of the remainder we can again cut out equal parts greater than half these remainders, and so on as far as we like. This process may be continued till the last remainder is smaller than any assignable quantity, however small. It follows, so we should conclude at present, that the two volumes must be equal, for they cannot differ by any assignable quantity.
To Greek mathematicians this conclusion offers far greater difficulties. They prove elaborately, by a reductio ad absurdum, that the volumes cannot be unequal. This proof must be read in the Elements. We must, however, state that we have in the above not proved Euclid's Prop. 5, but only a special case of it. Euclid does not suppose that the bases of the two pyramids to be compared are equal, and hence he proves that the volumes are as the bases. The reasoning of the proof becomes clearer in the special case, from which the general one may be easily deduced.
S 86. Prop. 6 extends the result to pyramids with polygonal bases. From these results follow again the rules at present given for the mensuration of solids, viz. a pyramid is the third part of a triangular prism having the same base and the same altitude. But a triangular prism is equal in volume to a parallelepiped which has the same base and altitude. Hence if B is the base and h the altitude, we have
Volume of prism = Bh, Volume of pyramid = 1/3Bh,
statements which have to be taken in the sense that B means the number of square units in the base, h the number of units of length in the altitude, or that B and h denote the numerical values of base and altitude.
S 87. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved surfaces:--
Prop. 10. Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.
Prop. 11. Cones or cylinders of the same altitude are to one another as their bases.
Prop. 12. Similar cones or cylinders have to one another the triplicate ratio of that which the diameters of their bases have.
Prop. 13. If a cylinder be cut by a plane parallel to its opposite planes or bases, it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other; which may also be stated thus:--
Cylinders on the same base are proportional to their altitudes.
Prop. 14. Cones or cylinders upon equal bases are to one another as their altitudes.
Prop. 15. The bases and altitudes of equal cones or cylinders are reciprocally proportional, and if the bases and altitudes be reciprocally proportional, the cones or cylinders are equal to one another.
These theorems again lead to formulae in mensuration, if we compare a cylinder with a prism having its base and altitude equal to the base and altitude of the cylinder. This may be done by the method of exhaustion. We get, then, the result that their bases are equal, and have, if B denotes the numerical value of the base, and h that of the altitude,
Volume of cylinder = Bh, Volume of cone = 1/3Bh.
S 88. The remaining propositions relate to circles and spheres. Of the sphere only one property is proved, viz.:--
Prop. 18. Spheres have to one another the triplicate ratio of that which their diameters have. The mensuration of the sphere, like that of the circle, the cylinder and the cone, had not been settled in the time of Euclid. It was done by Archimedes.