Mathematics simplified...

Front matter

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MATHEMATICS SIMPLIFIED

AND

CHAPTER II. Definitions of Mathematics

Chambers, Johnson, Eulcr, Comte, 29

Author's own Definition, 32

OfArithmetic 33

Author's Definition, 34

Of Geometry, - 35

Plato's criticism, 36

Aristotle. The relation of Geometry to Arithmetic, ... - 37

Author's Definition, 37

OfAlgebra, 40

Johnson, Newton, Chambers, Stewart, 41

Author's Definition, * 43

CHAPTER III. On the Nature of Demonstrative Evidence

Want of improved methods of instruction, 46

Quotations from Dr. Beddoes and Prof. Gilbert, 50

From Locke, 53

CHAPTER IV. Explanation of the Plates

The three great divisions of measurement, 55

Measurement — Linear, Superficial, Massive, 56

The relation of Elementary Geometry to the Laws of Motion, - - .58

v

VI CONTENTS.

The Parallelogram and the Triangle, and their relation to each other, - 59

PLATE No. 3.

Length and Breadth, and their relation to Fluent Quantities. Primary

idea of Fluxions, Analysis of Infinities, 61

Aggregates of Triangles. The Triangle, Square, Pentagon, Hexagon,

Octagon, and their approach to the Polygon, ... - . 64

The relation of the Polygon to the circle, that is, " squaring the circle," - 66 The Method of Exhaustions of Archimedes. There is or can be none other. 67 Squaring of the circle unmystified, and set at rest for ever, or a plain, intelligible, primary, and popular explanation of the Integral and Differential Calculus, - -69

PLATE No. 6.

Adjunct and subordinate to Plate No. 5, further illustrating differential

approximation, 71

PLATE No. 7.

The measurement of all plane figures, whether bounded by straight or

curve lines, is simply the result of Length and Breadth, 72

Description of the Plate, - 73

Application of Geometry to Mechanics, 75

Definition of Pure Mathematics, 76

Explanation of the Lever, in all its varieties, 78

The Wheel and Axle, the Pulley, the Balance, the Steelyard, - - - 79

Natural Philosophy — Light — Perspective, 81

Parallax — Celestial perspective, -------- 82

Heat, Electricity, Sound, Gravitation. All radiant phenomena illustrate

each other, 82

Same subjects continued, 83

Familiar illustration of Light and Heat, (" squares of the distances,") - 84

Gravity, exemplification of 86

Laws of forces, to be observed in their results, Motion, &c. ... 86

Centripetal and Centrifugal forces, 87

Reference to Plate No. 5, 88

Application to the orbits of the Planets and Comets, 89

Kepler. Retrospect of the previous progress of Astronomy. Pythagoras.

Copernicus. 90

TychoBrahe, 91

Hipparchus. Tycho. Bradley. --,,---- 92

Kepler. Tycho. Galileo 93

Kepler. The ellipticity of the orbit of Mars. His first law. - - 94 The Second Law of Kepler. The Radius Vector. Equal areas in equal

times, 95

The Pendulum. Galileo, 97

Notice of Galileo's researches, 98

The Third Law of Kepler, 101

Recapitulation of the Laws of Kepler, 102

Sir John HerschcPs eulogy of their surpassing importance, - - - 104

Confirmation of the Laws of Kepler, by Cassini & Newton, - - - 105 Their application by Cassini to the orbits of the satellites of Jupiter and

Saturn, 105

By Newton to the orbit of the great Comet of 1680, - - - - 106 Retrospect of Newton's earlier trains of thought in 1665-6, in relation to

the gravitation of the Moon, 107

Picard, in 1679, measures an arc of the meridian, and thus afforded Newton

correct data for determining the earth's radius, and enabled him to

apply the laws of Kepler to the earth's satellite, the Moon, - - 107

The comparative reciprocal attraction of the Planets, - - - - 108 Their comparative mutual disturbance of each other's motions in their

orbits determines their comparative weight, 109

The discovery of the new planet Neptune, 109

Mewton's generalization of universal gravitation, 110

Its application to the irregularities of the Moon's orbit, - » - - 111

Universal gravitation is but the generalization of the Laws of Kepler, - 112

PLATE No. 11.

A systematic series of diagrams, demonstrating the universality of tho

problem of Pythagoras, 112

VU1 CONTENTS.

PLATE No. 12 Further exemplification of the same problem, 115

Further exemplification of the same problem, which is capable of infinite

variety, 115

Pythagoras, 116

PLATES No. 14 & 15.

Further exposition of the same fundamental problem, - - - - 117 Logarithms, 118

PLATE No. 16.

Further exemplification of the same problem, with an Algebraic expression

which determines the Arithmetical relations of the series, - - - 118

The same series, with the addition of circles, beautifully exemplifying the application of this problem of Pythagoras to the Geometrical, Arithmetical, and Algebraic determination of rotary and radiant forces. The composition of two forces, geometrically exhibited, is but Length and Breadth. The ratio of diffusion of radiant phenomena is represented by Length, Breadth, and Thickness. It is the relation of the Cube, or of the sphere to its radius. - - . - . - - 121

Pythagoras, Archimides, Kepler, 121

PLATE No. 18.

Subordinate to Plate No. 17, in amplification of the same truths, - - 121

PLATE No. 19.

In further amplification of the same truths, which are capable of unlimited

application, 122

Poetical illustration of radiant phenomena. The Creation of Light, - 123 Concluding Remarks, 128

Preface

It must be acknowledged that by the existing methods, the science of Mathematics is seldom effectively and comprehensively taught. Much time is wasted on it with comparatively little good result to the great majority of students. It is the purpose of the present work to show that very great improvements can be made in the methods of instruction.

It may be thought that Mathematics is the last subject in which important improvement is to be expected, and that even an apology needs to be offered for an attempt so novel and audacious. In the opinion of many persons, the very cobwebs which have gathered around this science have acquired a time-honored and venerable antiquity, and he must be a bold man who dares to stir them.

If the history of the past affords a presage of the present, a storm of opposition may be expected to any great improvement in this subject, similar to that which, in Italy, assailed Galileo's revelation of the Solar System, and in England, during many years, suppressed Harvey's demonstration of the circulation of the blood.

On the other hand, it is certain that the existing methods of teaching fail to interest the great majority of minds ; that there are at this moment living, but few mathematicians in any comprehensive sense. However true, however ingenious, however worthy of immortal honor, the problems which the great Greek Geometers have bequeathed to us, as presented

b ix

PREFACE.

to the student in the pages of Euclid, these pages are to the great majority, emphatically tedious and repulsive ; they are very imperfectly comprehended, and therefore may practically be considered as little better than a sealed book; they are wanting in that simple, harmonious, and beautiful connection with each other, and with the glorious all-comprehensive superstructure of the physical laws, which would give them their highest value, and which has been reserved for a later age — an age subsequent to the labors of Galileo, of Tycho Brahe, and of Kepler.

Heretofore the appeal has been made so entirely to the memory rather than to the understanding, that we are not to be surprised at the repulsiveness of the methods by which the world has been so long mystified.

Mathematics, occupying as it does, a sort of intermediate position between physical science and mental philosophy, interwoven on the one hand with the forces and laws of Nature as well as of the Mechanic Arts, while, on the other hand as the science of absolute certainty, it is acknowledged to be the best means of cultivating a comprehensive and logical use of the intellectual faculties, is of indispensable importance to men of all professions ; to the scholar and to the man of science, to the logician and to the mechanic, to the artist and to the engineer, as well as to the naturalist and to the metaphysician.

In Mathematics, the human intellect is justly considered as having gradually accomplished its most perfect achievement, revealing the noblest evidences of the wisdom, omnipotence and omnipresence of the Creator. The highways to its

PREFACE. XI

grandest results have been discovered and investigated by men of the greatest genius — those noble martyrs of science, who, often amid the neglect or persecution of their cotemporaries, have passed their lives in patient devotion to their favorite study, have gloriously surmounted its rugged acclivities, and thus have entitled themselves to immortal gratitude and honor.

Though the compilation of Euclid was well for the age in which he lived, though it was a good text book twenty centuries ago, it may now fairly be assumed that the progress of the human mind, so conspicuous in every other department of industrial pursuits, and of physical science, imperatively demands for the nineteenth century a more efficient development of this great subject.

The truths of Mathematics, the laws of Nature, have existed unchanged. Two hundred generations of men have lived and died, ignorant of these laws, to an extent not creditable to the intellectual activity of our race.

Any well sustained attempt therefore to improve this science, and to facilitate and insure its general acquisition, deserves attention. It is honorable, even to attempt to simplify, to explain and to make attractive, a subject which ought to be universally understood and universally popular.

The art of Printing is certainly capable of achieving, and the demands of the age imperiously require a much better, as well as a much wider education among the teachers of the people and the people themselves.

Introduction

Mathematics is a science in which nothing can be taken upon authority. It is emphatically a science of demonstration, where self evident conviction alone, in the mind of each and every individual, is relied upon to command and to compel involuntary assent.

What is demonstration? A demonstration 'is a chain of axioms. An axiom is a self evident or admitted truth, a stubborn fact that cannot be disputed. Self evident truths are simple results of observation, experiment and experience. Thus the apparently most difficult demonstrations rest upon the simplest truths.

" To Demonstrate, verb active, [demonstro Latin,] to prove with the highest degree of certainty ; to prove in such a manner as reduces the contrary position to evident absurdity.

"Demonstration, noun substantive, [demonstratio, Latin,] the highest degree of deducible or argumental evidence ; the strongest degree of proof ; such proof as not only evinces the position proved to be true, but shows the contrary to be absurd and impossible." Johnson's Folio Die. 1755.

Words are the signs of ideas. Things, on the contrary, are the sources of ideas. An idea is a mental image ; an impression formed in the mind. An idea is an abstraction, when the source of it is not present. Thus, the idea or recollection of something that is past, is an abstraction. Thus, the general idea of justice, of truth, of natural philosophy, of mathematics, formed upon larger experience and education, is abstraction ; it is generalization. But an abstraction not founded upon, and not consonant with Nature and Truth, would be a falsity, an insanity.

The child in his earliest progress in education, has every thing to learn by observation and experience. He learns not only that one finger and one finger make two fingers, but that of any one thing added to another similar thing, (one unit added to another unit) make two, &c. He thus has begun to acquire abstract ideas. An idea is a likeness. "Abstract" means, taken from. Every abstract idea must have been preceded by a reality submitted to his senses, and more especially to his sight. An abstraction is a likeness. A likeness of what 1 Of something you have previously seen or known.

Yon Humboldt has said that no two persons assign pre. cisely the same meaning to a word. The significance which is attached by each individual to any conventional language, is necessarily a result of his previous knowledge or education. All conventional language has for its base a natural language, a language of signs, a pantomimic language.

The meaning of the simplest word which a child learns, is a result of observation or of education. The Abbe Sicard, in his researches into the origin of language in connection with the means of teaching the deaf and dumb, has, I think, established this fact. The word horse must be associated with the actual animal ; the word kindness must be associated with the natural expression of the emotion, before it can convey any meaning to the mind. The child first learns to discriminate a horse from a cow, and to associate the conventional words ; he very gradually becomes, by experience, acquainted with a considerable number of animals, and thus is capable of making more general comparisons ; in time he begins to observe their anatomy and comparative anatomy, and becomes an acute and philosophical observer. The idea of general classification at length occurs to the youthful Cuvier, and finally, in mature age, he writes his immortal work on the animal kingdom.

In like manner, the very simplest ideas of number, even that one and one make two, and that two and two make four, have to be acquired. A child, or an untaught savage may have a correct idea of numbers as far as he can count his fingers, but not beyond. It requires considerable familiarity with numbers to have an idea of hundreds, of thousands, or of millions, though we speak of them easily and thoughtlessly. Thus, the velocity of light, twelve millions of miles in one minute of time, though the fact that it is so, can be satisfactorily demonstrated, is too vast to be understood by uneducated persons without previous appropriate training.

In a science which professes to be so clear and certain as Mathematics, it is almost incredible that the use of terms should be so vague and unintelligible as we find it at this advanced period of the nineteenth century. There has evidently been too much assumption of profound meaning and too little honest intention to make intelligible to those of plain intellect and of common advantages, the truths of this noble and delightful science.

It is to this cause probably that a very large part of the difficulties of its acquirement is to be attributed, and that the nomenclature of the most certain of sciences is so little understood in an age like ours, boastfully claiming for itself an advanced intellectual and scientific culture.

Lord Brougham has lately said in the English Parliament, " that there is not a schoolmaster in England who has not himself need to go to school ;" and though the remark may not be very flattering to the professors of the age, it is probably true in a much wider and more emphatic sense even than its author intended.

In this country we have a class of public men who make it their ambition to be considered friends of education, and who are always ready to flatter us with the idea that our system of education is in a most flourishing condition. But with all this boasting it may be gravely doubted whether, relatively to the population, the number of men of really sound and comprehensive education is equal to what it was half a century ago.

Lord Bacon, in his much celebrated work, the "Novum Organon, or Rules for conducting the Understanding in search of Truth," published in 1620, Aphorism 90, says :

" In the customs and institutions of schools, universities, colleges, and the like conventions, destined for the seats of learned men, and the promotion of knowledge, all things are found opposite to the advancement of the sciences ; for the readings and exercises are here so managed, that it cannot easily come into any one's mind to think of things out of the common road. Or if here and there one should venture to use a liberty of judging, he can only impose' the task upon himself, without obtaining assistance from his fellows ; and if he could dispense with this, he will still find his industry and resolution a great hindrance to the raising of his fortune. For the studies of men in such places are confined and pinned down to the writings of certain authors, from which, if any man happens to differ, he is presently reprehended as a disturber and innovator. But there is surely a great difference between arts and civil affairs ; for the danger is not the same from new light, as from new commotions. In civil affairs; it is true, a change even for the better is suspected, through fear of disturbance; because these affairs depend upon authority, consent, reputation and opinion, and not upon demonstration. But arts and sciences should be like mines, resounding on all sides with new works and farther progress. And thus it ought to be, according to right reason ; but the case, in fact, is quite otherwise ; for the above-mentioned administration and policy of schools and universities, generally opposes and greatly prevents the improvement of the sciences."

Baden Powell, the present distinguished Professor of Mathematics in the university of Oxford, in his " History of Natural Philosophy" published in 1842, quotes and endorses this aphorism in the following paragraph :

" Bacon in proposing his new scheme of a philosophical institution, is of necessity led to justify the proposal by a reference to the services of existing institutions, and by showing how little they had done, or by their very nature and constitution were likely to do towards the real advancement of science. He censures, with severity indeed, but without bitterness, in strong terms, but with a masterly exposition of the facts, which evinces the perfect justice of his condemnation, the system of the colleges and universities of his day. [Nov. Organon, 1 Aph. 90.] He observes that the lectures and exercises were all of such a nature that no deviation from the established routine was likely to be thought of; that if a solitary attempt were made, the whole burden of it would rest on the individual, who would, moreover, find such attempts a serious impediment to his own advancement. The studies of these places were confined to a certain set of authors, as it were, imprisoned within those limits ; any one who should venture to deviate from this course, would be immediately condemned as a turbulent innovator. But, he adds, in the arts and sciences, as in mines, the whole region ought to resound with new works and further advances. Such was the state of the case in his day ; nor shall we find it much better in later times ; for the present it must suffice to remark that, in regard to Oxford, when the scholastic forms were in great measure broken up under the reign of Cromwell, the favorable opportunity which was afforded for establishing a better system in their place was, as we have seen, not lost by a few .ardent friends of true science ; owing however to a variety of causes, their partial attempts failed, as far as the university

c was concerned, and with the return of the Stuarts, the old system was re-established in all its authority."

Jared Sparks, on assuming the Presidency of Harvard University, June 20, 1849, says :

" Such have been the prodigious discoveries in the sciences, that the circle of what is called a liberal education has been greatly enlarged since the days of our fathers. The plan of instruction must be expanded in due proportions, and it is a problem difficult even in theory, but much more so in practice, how far one branch shall be curtailed to make way for another, which of them shall have the preference, and in what order the whole si. all be so adjusted that each may fall into its appropriate place, and contribute its appropriate share. If you appeal to the general voice, you will be met by a conflict of opinions that would multiply rather than clear away the difficulties. Each man would judge according to his own ideas of the relative importance of particular studies ; ideas derived rather from individual predilections, than from a comprehensive view of the whole subject. Some value lightly every kind of study which does not bear the stamp of direct utility. Others would have the thinking and reasoning faculties mainly cultivated ; while others again would recommend, as the highest achievement, the art of communicating thoughts by a mastery over language, so as to captivate and convince a listening multitude, or impress great truths and brilliant conceptions on the reading world. # # # #

" There is so general an opinion among students, particularly in their early college life, that the mysteries of the Mathematics are above their comprehension, the mathematical faculty being, as they suppose, a gift of nature conferred on a few favored mortals only, that it might be hazardous to deny a position maintained by such numbers, although it may be doubted whether it is the result of resolute and laborious experiment on their part. To accommodate such misgiving minds, however, it may not be amiss to leave Mathematics, at the proper stage of the college course, in the category of elective studies." #*##**

The distinguished and learned gentleman having thus briefly and coldly dismissed the study of mathematics, proceeds to pass a very strong, just, and eloquent encomium on the value of the study of the ancient classics.

" For the last thousand years, at all times, and in all countries, where learning of any sort has been thought worthy of attention, under every form of government, and at every stage of civilization, the ancient classics have made a prominent part in every system of education. It is a significant fact, also, that the numerous colleges in our extended republic it is believed, have followed this example. # # #

" The highest attainment of a man of action and of thought is the art of using language with accuracy, elegance, force, and effect." ######

True ! But how is the man of action and the man of thought to originate 1 How are boys to be trained, so as to become men of thought ? is the question.

It must be admitted that the mere art of using language is not necessarily indicative of valuable thought. And there is also another side to this question, as it is often possible to teach acknowledged idiots to talk ; as it is possible to teach persons of various grades of intellect above idiocy, to talk flippantly, fluently, and foolishly, so it is possible to teach men of higher grades of intellect, and even of naturally excellent abilities, to make talking in an ostentatious and pedantic style a habit and an ambition, and thus immensely to overrate themselves, and their true value to society.

It ought to be taught, and generally acknowledged, that to listen well, to know when to listen, and when to speak, is one of the highest accomplishments of a disciplined intellect. We want schools like that of Pythagoras, where students may go through a regular probationary course of silent listening and observation.

" The first step toward being wise is to know that we are ignorant." But most of us have not yet made even this first advance. Let us quote Locke, the greatest of mental philosophers :

" If we would speak of things as they are, we must allow that all the art of rhetoric, besides order and clearness, all the artificial and figurative application of words eloquence hath invented, are for nothing else but to insinuate wrong ideas, move the passions, and thereby mislead the judgment, and so, indeed, are a perfect cheat : and, therefore, however laudable or allowable oratory may render them in harangues, and popular addresses, they are certainly, in all discourses that pretend to inform and instruct, wholly to be avoided; and where truth and knowledge are concerned, cannot but be thought a great fault, either of the language, or person that makes use of them. What and how various they are, will be superfluous here to take notice : the books of rhetoric which abound in the world will instruct those who want to be informed ; only I cannot but observe how little the preservation and improvement of truth and knowledge is the care and concern of mankind, since the arts of fallacy are endowed and preferred. 'Tis evident how much men love to deceive and be deceived, since rhetoric, that powerful instrument of error and deceit, has its established professors, is publicly taught, and has always been had in great reputation ; and I doubt not but it will be thought great boldness, if not brutality, in me to have said thus much against it."

It is one of the evils of our age and country, some parts of it especially, for talkers, men whose chief ambition it is to hear themselves talk, to usurp a position to which their thinking capacities are utterly disproportioned. The business of our legislative halls, our legal tribunals, our town meetings, is often impeded by this rhetorical verbiage, to an extent which demands abatement.

True eloquence is nature; it is the natural expression of thought, which art but attempts to counterfeit.

A skilful use of language, a comprehensive knowledge of things, and a wise and reflective sagacity in discriminating the relations of causes and effects, so far from being antagonistic or opposite, are in truth intimately subservient and essential to each other ; and as an accomplished education is composed of an aggregate of many acquirements, whatever really facilitates, and thus saves time in the acquisition of one study, undoubtedly promotes the whole.

It is not only a false, but a very narrow and injurious doctrine, which supposes that a knowledge of those immortal discoveries of the Greek geometers, which form the basis of modern science, should excite in any degree less interest for those other forms in w T hich Grecian intellect has embodied itself, for the veneration of posterity.

Would it not be absurd to suppose that a knowledge of the Greek language, which is undoubtedly the most brilliant achievement of human genius, and the most remarkable memorial of the earliest real civilization, should diminish the estimation of Greek sculpture or Greek architecture 1 yet this would not be more absurd than for this knowledge to diminish in the mind of one who understands them, the value of the geometrical discoveries of Pythagoras, " the wisest of the Greeks."

Chapter I. On The Usefulness And Importance Of Mathematics

The usefulness of Mathematics is twofold. The first consists in its universal relation to physical science. The second in its relation to mental philosophy. It is perhaps equally valuable in both respects, as essential to every liberal and useful system of education.

It is important to men of every profession, as the means by which alone they are capable of understanding those principles which form the basis of all physical science — of all the physical laws of Nature. It may be said to form the framework of all the Sciences, and of all the Arts, or at least of all those of which Mechanics or Natural Philosophy constitute a part.

It is also, as the science in which our perceptions, reasonings, and conclusions are the clearest and least subject to doubt or dispute ; that in which our most important mental habits are formed and exercised, and in the prosecution of which steady and continued thought is indispensable, acknowledged to be the best means of cultivating a logical and comprehensive use of our intellectual faculties. Mathematics ought to be taught faithfully, and then they would be learned with delight by all persons who are ambitious of a good and sound education.

Instead, however, of extending his own views on the subject, the author begs to offer the following quotations from acknowledged authorities :

" Would you have a man reason well, you must use him to it betimes, exercise his mind in observing the connection of ideas, and follow them in train. Nothing does this better than mathematics, which, therefore, I think should be taught all

24 USEFULNESS AND IMPORTANCE

those who have the time and opportunity, not so much to make them mathematicians, as to make them reasonable creatures."

[Locke, on the Conduct of the Human Understanding, Introduction, Sec. 6.

" It is to be observed that as there is no study which may be entered upon with a less stock of preparatory knowledge than mathematics, so there is none in which a greater number of uneducated men have raised themselves by their own exertions to distinction and eminence."' — [Dean Swift.

" The mathematicians, from very plain and easy beginnings, by gentle degrees, and a continued chain of reasonings, proceed to the discovery and demonstration of truths that appear at first sight beyond human capacity." * _ # # # [Locke on the Improvement of our Knowledge, Bk. 4, ch. 12, s. 7.

" In mathematics, and in natural philosophy, since mathematics was applied to it, we see the noblest instances of the force of the human mind, and of the sublime heights to which it may rise by cultivation."

[Dugald Stewart, vol. 3, chap. 1, sec. 2.

" I have mentioned Mathematics as a way to settle in the mind a habit of reasoning closely, and in train; not that I think it necessary that all men should be deep mathematicians, but that having got the way of reasoning which that study necessarily brings the mind to, they might be able to transfer it to other parts of knowledge as they have occasion; for in all sorts of reasoning, every single argument should be managed as a mathematical demonstration, the connection and dependence of ideas should be followed, till the mind is brought to the source on which it bottoms, and observes the coherence all along." [Locke, Introduction to the conduct of the Human Under-