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T is my intention in this differtation, after saying something concerning the use of commentaries, to conduct the learner, step by step, to a folid, rational and extensive view of the principles of geometry; by explaining their original and improvement through several different gradations, until they put on a fcientific form; reducing what I have to say under distinct heads, for the benefit of the ignorant and thoughtless reader.
CHA P. I.
Of the neceffity and use of commentaries.
THE dulness of commentators is a fubject of fuch general complaint, that it may be proper to inquire how it comes to pass that books are not written in fuch a manner as to make a commentary useless or impertinent: and this inquiry may be proposed with the greater propriety, as the labours of commentators, befides their dulness, seldom or never answer the expectations of the public in other respects; which they no doubt are apt to imagine the author himself could have fully gratified, by condescending a little to the weakness of his readers, and rescued his works out of such clumfy hands.
What apology therefore shall I be able to make for loading with a commentary the most perfect book in the world! But though it may not be agreeable to that general delicacy which an author is bound to preserve towards his readers; yet it may nevertheless be proper to inform them, that they seem to be rather partial to themfelves upon the prefent queftion; never confidering how much of VOL. I.
the blame ought in all confcience to be laid at their own door: for
But I will even go farther and venture to affirm that an author, who writes upon fubjects of science, may find it often by no means convenient to deliver himself in fuch a manner as to be always intelligible even to thofe whom he would wish to have for readers. Because authors are confined to a particular method of arrangement, if it be their intention to deliver opinions or discoveries in a systematic manner. And although there has been a senseless and inceffant clamour, for almost two centuries, against this method; yet it seems to be the only way of keeping knowledge within such limits, as to be by any means manageable by the human mind.
It is true indeed that our progress in acquiring knowledge, when left to ourselves, and our own experience, is directly contrary to this, being first condemned to an examination of particulars. And even when we take up with the systems of others, it is requifite that we have laid in a fufficient ftock of materials to enable us readily to comprehend any very general doctrine. For no general rule, or law, or theorem, or what you please to call it, is by any means to be understood, unless we have particular inftances ready to apply, as occafion requires: always however excepting a genius of the kind given to Hudibras in the following lines, containing the meaning and importance of many volumes.
His notions fitted things fo well,
That which was which he could not tell.
The fault therefore is not in the systems, but arises from the general incapacity which mankind seem to labour under, for judging of the merits of fuch as have been offered to their confideration and as there are quacks of all denominations, ever lying in wait to take advantage of the fimplicity of the multitude; the world by this means has been over-run with counterfeits.
To have a genius for any science feems to me to imply a readinefs at finding out particular inftances to apply to any general rule. It is therefore to be fufpected that those who are deficient in this kind of invention will by no means find their progress in any science, answerable to the time which they bestow upon it: they may commit the rules or the theorems to memory and nevertheless be ignorant of their meaning and application. Hence a certain degree of invention becomes neceffary even for the ready acquisition of a fcience; and this perhaps not fo different from that kind of invention by which the principles were at first discovered, as many have been apt to imagine.
Now here is the difficulty; a fcientific book ought to contain knowledge in that compact form; in which every one would chuse to take a review of it, after he has made himself master of the subject and not incumbered with all those particular instances, which would now stand in the way of the readers imagination as much as they affifted it before. For instance, who could relish the nobleft of Newton's theorems if it presented itself to his mind encumbered with all those properties, which had led him from common notions to the right understanding of fo fublime a speculation.
A book therefore, if this reafoning be juft, which would be proper for a learner would be fit for nobody else: and the greatest perfection of writing will ftill leave occafion and employment for the talents of that kind of commentator, which I profefs myself to be. The sweepings of the author's ftudy would furnish the best and most authentic materials for works of this kind; and yet humble as the office may seem, if I can execute my task but nearly up to the idea which I have of it, I fhall not regret my labour. The reader however will be disappointed, if he expect to find Euclid either corrected or enlarged, my purpose being nothing more than
to conduct the learner to that fenfe in which the author wished himself to be understood. An oftentation of learning is a fault which prevails among many commentators; for they feem to be apprehensive leaft the world fhould think they know nothing but their business and therefore instead of explaining their author, endeavour to perfuade the world that they understand the subject better than he himself did. But I do not write to the ftrength but to the weakness of mankind; and therefore would not chuse to be confidered as challenging the whole world to find fault, but only as applying a remedy to a weakness which my own experience has found does exist.
CHA P. II.
Concerning the Original of the geometrical principles.
QUANTITY of both kinds, extenfion and number, is always forcing itself upon us whether we will or not; and must therefore leave fome very fixt and determinate impreffions upon the minds of the most inconfiderate. We cannot ftretch out our hand without receiving a perception of extenfion; nor open our eyes without feeing figured objects, bringing along with them to the mind a consciousness that they are more in number than one. This begets notions, which are by no means peculiar to any fingle art or profeffion, but common to all men. The geometrician felects the most accurate of these, and with fuch materials lays the foundation of his fcience. Particular circumstances have rendered some notions concerning quantity more invariable than others; or rather fo fixt that nothing can alter them. As it would difcover ridiculous affectation and ignorance to pretend to change these, which is indeed impoffible, fo it is alfo below the dignity of our author to affect to disguise them by any forced or unnatural conftruction, to make them wear a more philofophic appearance. But although the twelve common notions which he has felected, are to be understood according to the vulgar conception of them; yet the learner must give them a very particular examination: because it is not fufficient to have these notions, but we must also have the ready ufe of them