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T is my intention in this dissertation, after saying something

concerning the use of commentaries, to conduct the learner, step by step, to a solid, rational and extensive view of the principles of geometry; by explaining their original and improvement through several different gradations, until they put on a scientific form ; reducing what I have to say under distinct heads, for the benefit of the ignorant and thoughtless reader.

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THE dulness of commentators is a subject of such general complaint, that it may be proper to inquire how it comes to pass that books are not written in such 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, besides 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 clumsy hands.

What apology therefore shall I be able to make for loading with a commentary the must 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 themselves

upon the present question ; never considering how much of VOL. I.



the blame ought in all conscience to be laid at their own door: for though it may be a secret to a great many, yet it is an undoubted truth that the most perfect writer requires that his work should be put into the hands of those who can read. And my intention in this edition is not to correct my author, but to supply a defect which it could not have been very consistent with his plan to remedy. For he has written his book expressly upon the supposition that his reader was endued with the faculty of attention : and as this is a disposition of mind with which the book is but rarely taken up, though it will always be laid down with it, or else it has been used to little purpose : a few seasonable warnings therefore, to rouse the attention of the indolent reader, may be given with great propriety, and without bringing any reflexion upon the character of the author, who in point of perspicuity, excells every one.

But I will even go farther and venture to affirm that an author, who writes upon subjects of science, may find it often by no means convenient to deliver himself in such a manner as to be always intelligible even to those 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 incessant 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 requisite that we have laid in a sufficient stock 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 instances ready to apply, as occasion 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 so well,
That which was which he could not tell.


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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 such 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 seems to me to imply a readiness at finding out particular instances to apply to any general rule.


fieren a gun It is therefore to be suspected 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 *-*,,138 of invention becomes necessary even for the ready acquisition of science; and this perhaps not so 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 scientific 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 assisted it before. For instance, who could relish the noblest 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 so sublime a speculation.

A book therefore, if this reasoning be just, which would be proper for a learner would be fit for nobody else: and the greatest perfection of writing will ftill leave occasion and employment for the talents of that kind of commentator, which I profefs myself to be. The sweepings of the author's study would furnith 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 shall 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

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to conduct the learner to that sense in which the author wished himself to be understood. An oftentation of learning is a fault which prevails among many commentators ; for they seem to be apprehensive least the world should think they know nothing but their bufiness: and therefore instead of explaining their author, endeavour to persuade the world that they understand the subject better than he himself did. But I do not write to the strength but to the weakness of mankind; and therefore would not chuse to be considered 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.



Concerning the Original of the geometrical principles.

QUANTITY of both kinds, extension and number, is always forcing itself upon us whether we will or not; and must therefore leave fome very fixt and determinate impressions upon the minds of the most inconsiderate. We cannot stretch 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 single art or profeffion, but common to all men. The geometrician selects the most accurate of these, and with such materials lays the foundation of his science. Particular circumstances have rendered some notions concerning quantity more invariable than others; or rather so fixt that nothing can alter them. As it would discover ridiculous affectation and ignorance to pretend to change these, which is indeed impossible, so it is also below the dignity of our author to affect to disguise them by any forced or unnatural construction, to make them wear a more philofophic appearance. But although the twelve common notions which he has selected, 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 use of


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