The Analyst; or, a Discourse Addressed to an Infidel Mathematician

Front matter

Addreed to an

Infidel Mathematician.

It is examined whether the Object, Principles, and Inferences of the modern Analyis are more ditinctly conceived, or more evidently deduced, than Religious Myteries and Points of Faith.

Contents

SECT. I. Mathematicians preumed to be the great Maters of Reaon. Hence an undue deference to their deciions where they have no right to decide. This one Caue of Infidelity.

II. Their Principles and Methods to be examined with the ame freedom, which they aume with regard to the Principles and Myteries of Religion. In what Sene and how far Geometry is to be allowed an Improvement of the Mind.

III. Fluxions the great Object and Employment of the profound Geometricians in the preent Age. What thee Fluxions are.

I

HOUGH I am a Stranger to your Peron, yet I am not, Sir, a Stranger to the Reputation you have acquired, in that branch of Learning which hath been your peculiar Study; nor to the Authority that you therefore aume in things foreign to your Profeion, nor to the Abue that you, and too many more of the like Charader, are known to make of uch undue Authority, to the mileading of unwary Perons in matters of the highet Concernment, and whereof your mathematical Knowledge can by no means qualify you to be a competent Judge. Equity indeed and good Sene would incline one to diregard the Judgment of Men, in Points which they have not conidered or examined. But everal who make the loudet Claim to thoe Qualities, do, nevertheles, the very thing they would eem to depie, clothing themelves in the Livery of other Mens opinions, and putting on a general deference for the Judgment of you, Gentlement, who are preumed to be of all Men the greatet Maters of Reon, to be mot converant about ditinct Ideas, and never to take things upon trut, but always clearly to ee your way, as Men whoe contant Employment is the deducing Truth by the jutet inference from the mot evident Principles. With this bias on their Minds, they ubmit to your Deciions where you have no right to decide. And that this is one hort way of making Infidels I am credibly informed.

II

II. Whereas then it is uppoed, that you apprehend more ditinctly, conider more cloely, infer more jutly, conclude more accurately than other men, and that you are therefore les religious becaue more judicious, I hall claim the privilege of a Free-Thinker; and take the Liberty to inquire into the Object, Principles, and Method of Demontration admitted by the Mathematicians of the preent Age, with the ame freedom that you preume to treat the Principles and Myteries of Religion; to the end, that all Men may ee what right you have to lead, or what Encouragement others have to follow you. It hath been an older remark that Geometry is an excellent Logic. And it mut be owned, that when the Definitions are clear; when the Potulata cannot be refued, nor the Axioms denied; when from the ditinct Contemplation and Comparion of Figures, their Properties are derived, by a perpetual well-connected chain of Conequences, the Objects being till kept in view, and the attention ever fixed upon them; there is acquired an habit of reaoning, cloe and exact and methodical: which habit trengthens and harpens the Mind, and being transferred to other Subjects, is of general ue in the inquiry after Truth. But how far this is the cae of our Geometrical Analyts, it may be worth while to conider.

III

III. The Method of Fluxions is the general Key, by help whereof the modern Mathematicians unlock the ecrets of Geometry, and conequently of Nature. And as it is that which hath enabled them o remarkably to outgo the Ancients in dicovering Theorems and olving Problems, the exercie and application thereof is become the main, if not ole, employment of all thoe who in this Age pas for profound Geometers. But whether this Method be clear or obcure, conitent or repugnant, demontrative or precarious, as I hall inquire with the utmot impartiality, o I ubmit my inquiry to your own Judgment, and that of every candid Reader. Lines are uppoed to be generated by the motion of Points, Plains by the motion of Lines, and Solids by the motion of Plains. And whereas Quantities generated in equal times are greater or leer, according to the greater or leer Velocity, wherewith they increae and are generated, a Method hath been found to determine Quantities from the Velocities of their generating Motions.

And uch Velocities are called Fluxions: and the Quantities generated are called flowing Quantities. Thee Fluxions are aid to be nearly as the Increments of the flowing Quantities, generated in the leat equal Particles of time; and to be accurately in the firt Proportion of the nacent, or in the lat of the evanecent, Increments. Sometimes, intead of Velocities, the momentaneous Increments or Decrements of undetermined flowing Quantities are conidered, under the Appellation of Moments.

IV

IV. By Moments we are not to undertand finite Particles. Thee are aid not to be Moments, but Quantities generated from Moments, which lat are only the nacent Principles of finite Quantities. It is aid, that the minutet Errors are not to be neglected in Mathematics: that the Fluxions are Celerities, not proportional to the finite Increments though ever o mall; but only to the Moments or nacent Increments, whereof the Proportion alone, and not the Magnitude, is conidered. And of the aforeaid Fluxions

there be other Fluxions, which Fluxions of Fluxions are called econd Fluxions. And the Fluxions of thee econd Fluxions are called third Fluxions: and oon, fourth, fifth, ixth, &c. ad infinitum. Now as our Sene is trained and puzzled with the perception of Objects extremely minute, even o the Imagination, which Faculty derives from Sene, is very much trained and puzzled to frame clear Ideas of the leat Particles of time, or the leat Increments generated therein: and much more o to comprehend the Moments, or thoe Increments of the flowing Quantities in tatu nacenti, in their very firt origin or beginning to exit, before they become finite Particles. And it eems till more difficult, to conceive the abtracted Velocities of uch nacent imperfect Entities. But the Velocities of the Velocities, the econd, third, fourth and fifth Velocities, &c. exceed, if I mitake not, all Humane Underftanding. The further the Mind analyeth and purueth thee fugitive Ideas, the more it is lot and bewildered; the Objects, at firt fleeting and minute, oon vanihing out of ight. Cer

in any Sene a econd or third Fluxion eems an obcure Mytery. The incipient Celerity of an incipient Celerity, the nacent Augment of a nacent Augment, i. e. of a thing which hath no Magnitude: Take it in which light you pleae, the clear Conception of it will, if I mitake not, be found impoible, whether it be o or no I appeal to the trial of every thinking Reader. And if a econd Fluxion be inconceivable, what are we to think of third, fourth, fifth Fluxions, and o onward without end?

V

V. The foreign Mathematicians are uppoed by ome, even of our own, to proceed in a manner, les accurate perhaps and geometrical, yet more intelligible. Intead of flowing Quantities and their Fluxions, they conider the variable finite Quantities, as increaing or diminihing by the continual Addition or Subduction of infinitely mall Quantities. Intead of the Velocities wherewith Increments are generated, they conider the Increments or Decrements themelves, which they call Differences, and which are sup

to be infinitely mall. The Difference of a Line is an infinitely little Line; of a Plain an infinitely little Plain. They uppoe finite Quantities to conit of Parts infinitely little, and Curves to be Polygones, whereof the Sides are infinitely little, which by the Angles they make one with another determine the Curvity of the Line. Now to conceive a Quantity infinitely mall, that is, infinitely les than any enible or imaginable Quantity, or than any the leat finite Magnitude, is, I confes, above my Capacity. But to conceive a Part of uch infinitely mall Quantity, that hall be till infinitely les than it, and conequently though multiply'd infinitely hall never equal the minutet finite Quantity, is, I upect, an infinite Difficulty to any Man whatoever; and will be allowed uch by thoe who candidly ay what they think; provided they really think and reflect, and do not take things upon trut.

VI

VI. And yet in the calculus differentialis, which Method erves to all the ame Intents and Ends with that of Fluxions,

our modern Analyts are not content to conider only the Differences of finite Quantities: they alo conider the Differences of thoe Differences, and the Differences of the Differences of the firt Differences. And o on ad infinitum. That is, they conider Quantities infinitely les than the leat dicernible Quantity; and others infinitely les than thoe infinitely mall ones; and till others infinitely les than the preceding Infiniteimals, and o on without end pr limit. Inomuch that we are to admit an infinite ucceion of Infiniteimals, each infinitely les than the foregoing, and infinitely greater than the following. As there are firt, econd, third, fourth, fifth, &c. Fluxions, o there are Differences, firt, econd, third, fourth, &c. in an infinite Progreion towards nothing, which you till approach and never arrive at. And (which is mot trange) although you hould take a Million of Millions of thee Infiniteimals, each whereof is uppoed infinitely greater than ome other real Magnitude, and add them to the leat given Quantity, it hall be never the bigger. For this is one of the modet potulata of our modern Mathematicians, and is a Corner-tone or Ground-work of their Speculations.

VII

VII. All thee Points, I ay, are uppoed and believed by certain rigorous Exactors of Evidence in Religion, Men who pretend to believe no further than they can ee. That Men, who have been converant only about clear Points, hould with difficulty admit obcure ones might not eem altogether unaccountable. But he who can diget a econd or third Fluxion, a econd or third Difference, need not, methinks, be queamih about any Point in Divinity. There is a natural Preumption that Mens Faculties are made alike. It is on this Suppoition that they attempt to argue and convince one another. What, therefore, hall appear evidently impoible and repugnant to one, may be preumed the ame to another. But with what appearance of Reaon hall any Man preume to ay, that Myteries may not be Objects of Faith, at the ame time that he himelf admits uch obcure Myteries to be the Object of Science?

VIII

VIII. It mut indeed be acknowledged, the modern Mathematicians do not conider thee Points as Myteries, but as clearly conceived and matered by their comprehenive Minds. They cruple not to ay, that by the help of thee new Analytics they can penetrate into Infinity it elf: That they can even extend their Views beyond Infinity: that their Art comprehends not only Infinite, but Infinite of Infinite (as they expres it) or an Infinity of Infinites. But, notwithtanding all thee Aertions and Pretenions, it may be jutly quetioned whether, as other Men in other Inquiries are often deceived by Words or Terms, o they likewie are not wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species. Nothing is eaier than to devie Expreions or Notations for Fluxions and Infiniteimals of the firt, econd, third, fourth and ubequent Orders, proceeding in the ame regular form without end or limit &c. or dx. ddx. dddx. ddddx &c. Thee Expreions indeed are clear and ditinct, and the Mind finds no difficulty in conceiving them to be continued beyond any aignable Bounds. But if we remove the Veil and look underneath, if laying aide the Expreions we et our elves attentively to conider the things themelves, which are uppoed to be expreed or marked thereby, we hall dicover much Emptines, Darknes, and Confuion; nay, if I mitake not, direct Impoibilities and Contradictions. Whether this be the cae or no, every thinking Reader is intreated to examine and judge for himelf.

IX

IX. Having conidered the Object, I proceed to conider the Principles of this new Analyis by Momentums, Fluxions, or Infiniteimals; wherein if it hall appear that your capital Points, upon which the ret are uppoed to depend, include Error and fale Reaoning; it will then follow that you, who are at a los to conduct your elves, cannot with any decency et up for guides to other Men. The main Point in the method of Fluxions is to obtain the Fluxion or Momentum of the Rectangle or Product of two indeterminate Quantities. Inamuch as from thence are derived Rules for obtaining the Fluxions of all other Products and Powers; be the Coefficients or the Indexes what they will, integers or fractions, rational or urd. Now this fundamental Point one would think hould be very clearly made out, conidering how much is built upon it, and that its Influence extends throughout the whole Analyis. But let the Reader judge. This is given for Demontration. Suppoe the Product or Rectangle AB increaed by continual Motion: and that the momentaneous Increments of the Sides A and B are a and b. When the Sides A and B were deficient, or leer by one half of their Moments, the Rectangle was i.e. . And as oon as the Sides A and B are increaed by the other two halves of their Moments, the Rectangle becomes or . From the latter Rectangle ubduct the former, and the remaining difference will be aB + bA. Therefore the Increment of the Rectangle generated by the intire Increments a and b is aB + bA. Q.E.D. But it is plain that the direct and true Method to obtain the Moment or Increment of the Rectangle AB, is to take the Sides as increaed by their whole Increments, and o multiply them together, A + a by B + b, the Product whereof AB + aB + bA + ab is the augmented Rectangle whence if we ubduct AB, the Remainder aB + bA + ab will be the true Increment of the Rectangle, exceeding that which was obtained by the former illegitimate and indirect Method by the Quantity ab. And this holds univerally be the Quantities a and b what they will, big or little, Finite or Infiniteimal, Increments, Moments, or Velocities. Nor will it avail to ay that ab is a Quantity exceeding mall: Since we are told that in rebus mathematicis errores quàm minimi non unt contemnendi. Such reaoning as this, for Demontration, nothing but the obcurity of the Subject could have encouraged or induced the great Author of the Fluxionary Method to put upon his Followers, and nothing but an implicit deference to Authority could move them to admit. The Cae indeed is

difficult. There can be nothing done till you have got rid of the Quantity ab. In order to this the Notion of Fluxions is hifted: It is placed in various Lights: Points which hould be clear as firt Principles are puzzled' and Terms which hould be teadily ued are ambiguous. But notwithtanding all this addres and skill the point of getting rid of ab cannot be obtained by legitimate reaoning. If a Man by Methods, not geometrical or demontrative, hall have atisfied himelf of the uefulnes of certain Rules; which he afterwards hall propoe to his Diciples for undoubted Truths; which he undertakes to demontrate in a ubtile manner, and by the help of nice and intricate Notions; it is not hard to conceive that uch his Diciples may, to ave themelves the trouble of thinking, be inclined to confound the uefulnes of a Rule with the certainty of a Truth, and accept the one for the other; epecially if they are Men accutomed rather to compute than to think; earnet rather to go on fat and far, than olicitous to et out warily and ee their way ditinctly.

XI

XI. The Points or meer Limits of nacent Lines are undoubtedly equal, as having no more magnitude one than another, a Limit as uch being no Quantity. If by a Momentum you mean more than the very initial Limit, it mut be either a finite Quantity or an Infiniteimal. But all finite Quantities are exprely excluded from the Notion of a Momentum. Therefore the Momentum mut be an Infiniteimal. And indeed, though much Artifice hath been employed to ecape or avoid the admiion of Quantities infinitely mall, yet it eems ineffedtual. For ought I ee, you can admit no Quantity as a Medium between a finite Quantity and nothing, without admitting Infiniteimals. An Increment generated in a finite Particle of Time, is it elf a finite Particle; and cannot therefore be a Momentum. You mut therefore take an Infiniteimal Part of Time wherein to generate your Momentum. It is aid, the Magnitude of Moments is not conidered: And yet thee ame Moments are uppoed to be divided into Parts. This is not eay to conceive, no more than it is why we hould take

Quantities les than A and B in order to obtain the Increment of AB, of which proceeding it mut be owned the final Caue or Motive is very obvious; but it is not o obvious or eay to explain a jut and legitimate Reaon for it, or hew it to be Geometrical.

XII

XII. From the foregoing Principle o demontrated, the general Rule for finding the Fluxion of any Power of a flowing Quantity is derived. But, as there eems to have been ome inward Scruple or Conciounes of defect in the foregoing Demontration, and as this finding the Fluxion of a given Power is a Point of primary Importance, it hath therefore been judged proper to demontrate the ame in a different manner independent of the foregoing Demontration. But whether this other Method be more legitimate and concluive than the former, I proceed now to examine; and in order thereto hall premie the following Lemma. "If with a View to demontrate any Propoition, a certain Point is uppoed, by virtue of which certain other Points are attained; and uch uppoed Point be it elf afterwards detroyed or rejected by a contrary Suppoition; in that cae, all the other Points, attained thereby and conequent thereupon, mut alo be detroyed and rejected, o as from thence forward to be no more uppoed or applied in the Demontration." This is o plain as to need no Proof.

XIII

XIII. Now the other Method of obtaining a Rule to find the Fluxion of any Power is as follows. Let the Quantity x flow uniformly, and be it propoed to find the Fluxion of xn. In the ame time that x by flowing becomes x + o, the Power xn becomes , i. e. by the Method of infinite Series &c. and the Increments o and &c. are one to another as 1 to &c. Let now the Increments vanih, and their lat Proportion will be 1 to . But it hould eem that this reaoning is not fair or concluive. For when it is aid, let the Increments vanih, i. e. let the Increments be nothing, or let there be no Increments, the former Suppoition that the Increments were omething, or that there were Increments, is detroyed, and yet a Conequence of that Suppoition, i. e. an Expreion got by virtue thereof, is retained. Which, by the foregoing Lemma, is a fale way of reaoning. Certainly when we uppoe the Increments to vanih, we mut uppoe their Proportions, their Expreions, and every thing ele derived from the Suppoition of their Exitence to vanih with them.

XIV

XIV. To make this Point plainer, I hall unfold the reaoning, and propoe it in a fuller light to your View. It amounts therefore to this, or may in other Words be thus expreed. I uppoe that the Quantity x flows, and by flowing is increaed, and its Increment I call o, o that by flowing it becomes x + o. And as x increaeth, it follows that every Power of x is likewie increaed in a due Pro . Therefore as x becomes x + o, xn will become : that is, according to the Method of infinite Series, &c. And if from the two augmented Quantities we ubduct the Root and the Power repectively, we hall have remaining the two Increments, to wit, o and &c. which Increments, being both divided by the common Divior o, yield the Quotients 1 and &c. which are therefore Exponents of the Ratio of the Increments. Hitherto I have uppoed that x flows, that x hath a real Increment, that o is omething. And I have proceeded all along on that Suppoition, without which I hould not have been able to have made o much as one ingle Step. From that Suppoition it is that I get at the Increment of x^n, that I am able to compare it with the Increment of x, and that I find the Proportion between the two Increments. I now beg leave to make a new Suppoition contrary to the firt, i. e. I will uppoe that there is no Increment of x, or that o is nothing; which econd Suppoition detroys my firt, and is inconitent with it, and therefore with every thing that uppoeth it. I do nevertheles beg leave to retain , which is an Expreion obtained in virtue of my firt Suppoition, which necearily preuppoeth uch Suppoition, and which could not be obtained without it: All which eems a mot inconitent way of arguing, and uch as would not be allowed of in Divinity.

XV

XV. Nothing is plainer than that no just Concluion can be directly drawn from two inconitent Suppoitions. You may indeed uppoe any thing poible: But afterwards you may not uppoe any thing that detroys what you firt uppoed. Or if you do, you mut begin de novo. If therefore you uppoe that the Augments vanih, i. e. that there are no Augments, you are to begin again, and ee what follows from uch Suppoition. But nothing will follow to your purpoe. You cannot by that means ever arrive at your Concluion, or ucceed in, what is called by the celebrated Author, the Invetigation of the firt or lat Proportions of nacent and evanecent Quantities, by intituting the Analyis in finite ones. I repeat it again: You are at liberty to make any poible Suppoition: And you may detroy one Suppoition by another: But then you may not retain the Conequences, or any part of the Conequences of your firt Suppoition o detroyed. I admit that Signs may be made to denote either any thing or nothing: And conequently that in the original Notation x + o, o might have ignified either an Increment or nothing. But then which of thee oever you make it ignify, you mut argue conitently with uch its Signification, and not proceed upon a double Meaning: Which to do were a manifet Sophim. Whether you argue in Symbols or in Words, the Rules of right Reaon are till the ame. Nor can it be uppoed, you will plead a Privilege in Mathematics to be exempt from them.

XVI

XVI. If you aume at firt a Quantity increaed by nothing, and in the Expres x + o, tands for nothing, upon this Suppoition as there is no Increment of the Root, o there will be no Increment of the Power; and conequently there will be none except the firt, of all thoe Members of the Series contituting the Power of the Binomial; you will therefore never come at your Expreion of a Fluxion legitimately by uch Method. Hence you are driven into the fallacious way of proceeding to a certain Point on the Suppoition of an Increment, and then at once hifting your Suppoition to that of no Increment. There may eem great Skill in doing this at a certain Point or Period. Since if this econd Suppoition had been made before the common Diviion by o, all had vanihed at once, and you mut have got nothing by your Suppoition. Whereas by this Artifice of firt dividing, and then changing your Suppoition, you retain 1 and , notwithtanding all this addres to cover it, the fallacy is till the ame. For whether it be done ooner or later, when once the econd Suppoition or Aumption is made, in the ame intant the former Assumpti and all that you got by it is detroyed, and goes out together. And this is univerally true, be the Subjectt what it will, throughout all the Branches of humane Knowledge; in any other of which, I believe. Men would hardly admit uch a reaoning as this, which in Mathematics is accepted for Demontration.

XVII

XVII. It may not be amis to oberve, that the Method for finding the Fluxion of a Rectangle of two flowing Quantities, as it is et forth in the Treatie of Quadratures, differs from the abovementioned taken from the econd Book of the Principles, and is in effect the ame with that ued in the calculus differentialis. For the uppoing a Quantity infinitely diminihied and therefore rejecting it, is in effect the rejecting an Infiniteimal; and indeed it requires a marvellous harpnes of Dicernment, to be able to ditinguih between evanecent Increments and infiniteimal Differences. It may perhaps be aid that the Quantity being infinitely diminished becomes nothing, and o nothing is rejected. But according to the received Principles it is evident, that no Geometrical Quantity, can by any diviion or ubdiviion whatoever be exhauted, or reduced to nothing. Conidering the various Arts and Devices ued by the great Author of the Fluxionary Method: in how many Lights he placeth his Fluxions: and in what different ways he attempts to demontrate the ame Point: one would be inclined to think, he was himelf upicious of the jutnes of his own demontrations; and that he was not enough pleaed with any one notion teadily to adhere to it. Thus much at leat is plain, that he owned himelf atisfied concerning certain Points, which nevertheles he could not undertake to demontrate to others. Whether this atisfaction aroe from tentative Methods or Inductions; which have often been admitted by Mathematicians, (for inftance by Dr. Wallis in his Arithmetic of Infinites) is what I hall not pretend to determine. But, whatever the Cae might have been with repect to the Author, it appears that his Followers have hewn themelves more eager in ap his Method, than accurate in examining his Principles.

XVIII

XVIII. It is curious to oberve, what ubtilty and skill this great Genius employs to truggle with an inuperable Difficulty; and through what Labyrinths he endeavours to ecape the Doctrine of Infiniteimals; which as it intrudes upon him whether he will or no, o it is admitted and embraced by others without the leat repugnance. Leibnitz and his Followers in their calculus differentialis making no manner of cruple, firt to uppoe, and econdly to reject Quantities infinitely mall: with what clearnes in the Apprehenion and jutnes in the reaoning, any thinking Man, who is not prejudiced in favour of thoe things, may eaily dicern. The Notion or Idea of an infiniteimal Quantity, as it is an Object imply apprehended by the Mind, hath been already conidered. I hall now only oberve as to the method of getting rid of uch Quantities, that it is done without the leat Ceremony. As in Fluxions the Point of firt importance, and which paves the way to the ret, is to find the Fluxion of a Product of two indeterminate Quantities, o in the calculus differentialis (which Method is uppoed to have been borrowed from the former with ome mall Alterations) the main Point is to obtain the difference of uch Product. Now the Rule for this is got by rejecting the Product or Rectangle of the Differences. And in general it is uppoed, that no Quantity is bigger or leer for the Addition or Subduction of its Infiniteimal: and that conequently no error can arie from uch rejection of Infiniteimals.

XIX

XIX. And yet it hould eem that, whatever errors are admitted in the Premies, proportional errors ought to be apprehended in the Concluion, be they finite or infiniteimal: and that therefore the ἀχρίβεια of Geometry requires nothing hould be neglected or rejected. In anwer to this you will perhaps ay, that the Concluions are accurately true, and that therefore the Principles and Methods from whence they are derived mut be o too. But this inverted way of demontrating your Principles by your Concluions, as it would be peculiar to you Gentlemen, o it is contrary to the Rules of Logic. The truth of the Concluion will not prove either the Form or the Matter of a Syllogim to be true: inamuch as the Illation might have been wrong or the Premies fale, and the Concluion nevertheles true, though not in virtue of uch Illation or of uch Premies. I ay that in every other Science Men prove their Concluions by their Principles, and not their Principles by the Concluions. But if in yours you hould allow your elves this unnatural way of proceeding, the Conequence would be that you mut take up with the Induction, and bid adieu to Demontration. And if you ubmit to this, your Authority will no longer lead the way in Points of Reaon and Science.

XX

XX. I have no Controvery about your Concluions, but only about your Logic and Method. How you demontrate? What Objects you are converant with, and whether you conceive them clearly? What Principles you proceed upon; how ound they may be; and how you apply them? It mut be remembred that I am not concerned about the truth of your Theorems, but only about the way of coming at them; whether it be legitimate or illegitimate, clear or obcure, cientific or tentative. To prevent all poibility of your mitaking me, I beg leave to repeat and init, that I conider the Geometrical Analyt as a Logician, i. e. o far forth as he reaons and argues; and his Mathematical Concluions, not in themelves, but in their Premifes; not as true or fale, ueful or inignificant, but as derived from uch Principles, and by uch Inferences. And foramuch as it may perhaps eem an unaccountable Paradox, that Mathematicians hould deduce true Propoitions from fale Principles, be right in the Concluion, and yet err in the Premies; I hall endeavour particularly to explain why this may come to pas, and hew how Error may bring forth Truth, though it cannot bring forth Science.