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As the knowledge I have acquired, in fuch ftudies, has been entirely from Books, without any other affiftance; I may, perhaps, have been able to fee deficiences or fuperfluities in them, better than those who have ftudied under a Tutor; and, I dare venture to affirm, that very few, who are not already tolerably verfed in Geometry, will be able to form any Idea of what the 7th Propofition, Book I. attempts to prove; who, from infpection only of a proper Figure, might be fully convinced of the truth of it.

I have, in this Treatife, endeavoured to render every Propofition eafy and intelligible, to any capacity; and have omitted none that are ufeful, or neceffary to demonftrate the reft; and, I will be bold to affirm, that those who cannot, from it, acquire a knowledge of all that is requifite, in Plane Geometry, will never be able to attain it at all. Where there is not a capacity to understand, they had better defift from the undertaking. An equal talent is not given to all; and, for fuch as have not a talent and a natural propenfity, to expect to attain any tolerable share of knowledge, in Geometry, is aiming at impoffibilities.

The 16th and 17th Propofitions, Bock 1ft, are entirely useless; for fince, in the 32nd, the external Angle is proved to be equal to the two remote Angles of the Triangle; and, that all the three Angles, of every Triangle, are equal to two right ones; it feems abfurd to prove, before hand, that any two of the Angles are lefs than two right Angles, and, that the external Angle, is greater than either of the remote ones. As if one fhould undertake, first, to prove that five is greater than either two, or three, and afterwards, that it is equal to them both. I have, therefore, made free, to alter, in fome meafure, the Elements of the firft Book; that, by a different arrangement, in tranfpofing the places

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of fome Propofitions, others, which depend on them, may with more eafe and elegance be demonftrated.

I fhall ever be of opinion with Tacquet, and fome others, that, to attempt a formal Demonftration of Propofitions which are felf-evident, is involving a thing, in itfelf clear and confpicuous, in darknefs and obfcurity. I have always found more difficulty in demonftrating, to another Perfon, felf-evident Propofitions, than the most intricate of others; and, when done, have only confused the Idea which the Pupil had more perfectly from infpection of the Figure. If a knowledge of feveral properties of Figures be acquired, which is neceffary to elucidate other more fublime Propofitions, is not the eafieft method the most eligible? certainly it is; and, barely to be told feveral properties of Triangles is fufficient for attaining the reft. Therefore, in this Work, I have reduced fome Propofitions into Corollaries; as they are but a certain confequence of the preceding Propofition, or of fome other.

Dr. Keill, in his Preface to his Tranflation of Commandine, and alfo Mr. Cunn feems to think it an unpardonable fault in Tacquet, to omit the Demonftrations of the 5th Book; and afferts, that not one Demonftration, in the 6th, 11th and 12th, can be obtained without it. But, I must beg his pardon, for differing from him in opinion, and am, myfelf, a living witness against. fuch his affertion; for, I do aver, that, without any other demonstration of the 5th Book, than what Tacquet has delivered, I have been able to go through all the other; and, without vanity, I think I understand them; but, of that, let this Treatife bear witnefs. I never could, at firft, have had patience to go through the dry and tedious Demonftrations, delivered in 25 Propofitions, of the 5th Book of his Euclid; and, had it firft fallen in my way, I fhould certainly have lain it afide before I had got through a fourth part of it; yet, I must acknowledge, that Tacquet is as much too brief as the other is tedious.

I cannot think it abfolutely neceffary, in order to obtain a competent knowledge in Geometry, and of Proportion in particular, to tread exactly in the fame fteps with Euclid; therefore, I have made

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made free to deviate, where I think it for the better, a readier or clearer way of investigating. I have, likewise, separated the Problems from the Theorems; making, of them, a diftinct and feparate Book, for the ufe of Schools and for Mechanics in general; the full Demonftration, of which, may be obtained from the Elements, to which I refer where it is neceffary. I have given it the first place, contrary to fome others who have made it the last Book, or otherwife difpofed of it. By which means, we are frequently at a lofs in the References, and are told to form Confructions, before we have learned how.

My reafon for feparating Practical Geometry from the Theorems is, that there are numbers of People, to whom it may be of great ufe, who either have not leisure or inclination to go through, or, perhaps, a capacity to understand the Demonftrations of the previous Propofitions, which difcourages them from proceeding. But, fuch as are not inclined to take it on credit, will readily obtain the Demonftration from the Theorems.

To make the Work more compleat. I have added, after Practical Geometry, a brief Theory of the nature and conftruction of an Ellipfis, that most useful and valuable Figure. And also, as an Appendix to the whole, I have given a concife Theory of menfuration of Superficies and Solids; fhewing their immediate and abfolute dependence on Geometry.

I have for fome time debated, with myfelf, whether I fhould publish a tract of Geometry or not; from which I have been deterred, through the perfuafions of fome particular Acquaintance; alledging, that there are more learned Treatifes already published, than any I could produce: the truth of which, I readily affent to; but, I don't find that they are more intelligible for that, and have, therefore, determined to publish. My chief reafon, for which, is to bring the whole Elements into lefs compafs, and to abridge it of that tedious prolixity, which is in many Authors, on that Subject; in dwelling too long on fuch Propofitions as are clear and evident of themselves; and, by that means, to render the ftudy of Geometry more pleasant and entertaining. 'Tis enough to deter any one from the purfuit of a Science, who find, in the Rudiments of it,

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fo much perplexity; for, I am perfuaded, that, uniefs a Perfon has a ftrong inclination to it, and a good natural Capacity, 'tis not a very pleafing Study, at the firft, until they begin to feel the fublime Truths it contains; and am therefore of opinion, that, the easier it is made in the begining, the more entertaining to young, Students.

Can there need Demonftration, that any two fides of a Triangle, are greater than the third? is there a Perfon fo ignorant as not to know it? it is implanted in us by Nature; every common Porter knows it, or practices it every Day. Who ever faw one of them traverse two fides of a Square, when he could cross the Diagonal? and why is it? but, because he knows it to be shorter than the two Sides. Is it not obvious, that the greateft Angle of every Triangle, must be oppofite to the greatest Side ? and can there be any need to demonftrate the converfe? that the greater Side fubtends, or is oppofite to the greater Angle; is not the one contained in the other? it is trifling, to no purpose; for, all converfe Propofitions may be Corollaries to the former. If two Sides of a Triangle, equal to two Sides of another Triangle, contain a greater or lefs Angle, the Bafe will be greater or lefs? Are not all thefe, and feveral more, obvious and clear, from a bare infpection of the Figure? nay even without it; 'tis enough to be told they have fuch properties, and not to lofe time in trying the patience of the Student, with a tedious and puzling Demonftration of what he faw clearer before; for, if the thing is feen or known, what needs there more? is it intended to perplex, only, where it can be of no use? to difguft the beginner, before he is able to fee any of the Beauties it contains? Yet, I do not omit thefe entirely, because the whole Elements depend on them; but have endeavoured to treat them in as fimple a manner as the nature of the Subject will admit of; if I have been concife, I doubt not I fhall be excufed, if I have but faid enough.

But, as I think I have faid enough, in this place, Ifhall ftraightway proceed to the Subject; through which, if the Reader be inclined to follow, with an intention to learn what it contains, I am

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much miftaken if he lofes his labour; and, for fuch as perufe it only with intent to cavil, I hope they will be greatly difappointed, and find but little to cavil at. I am not fo vain as to fuppofe it is without defect, or that it will please all, for that I know is impoffible; but, if I have made it intelligible and useful to common capacities, it is what I aimed at, and fhall reft fatisfied in the fuppofition that my labour is not entirely loft.

The greatest fault in Tacquet is, that his Figures are too small and trivial; and it is a general fault, that they are often incorrect and frequently contradict the defcription. Correct and well adapted Schemes are certainly of fome confequence, in which, I have been very particular; and have, alfo, carefully revifed the letter prefs, fo that, I hope there are but few errors have efcaped my obfervation. For, Errors, in mifplacing the Referrences to the Schemes, and fometimes omitting them entirely (which is better of the two) is unpardonable in mathematical Works; having frequently experienced, in most Authors that I have perused, the perplexity it occafions; especially, when the subject is quite new to the Student. But, if any fhould remain unnoticed, I hope the candid reader will impute it to human fallibility, and not quarrel with it on that account, for I am certain he will not meet with many.

Although I have, in this Treatife, deviated greatly from Euclid, in many particulars, I have endeavoured to make it generally ufeful; by putting his Numbers after mine, to each Propofition, and also by means of an Index, I have fhewn where to find any Propofition of Euclid; which, in cafe of reference, to Euclid, in other Works, may be readily turned to.

I have well confidered and digefted every Propofition, have carefully revised them over and over with the ftrictest attention, and I am fully convinced, that there is no omiffion of any thing that is effential, or neceffary to be known. Notwithstanding, I have abridged the whole Elements, particularly the first, the third, the fifth, and the eleventh Books; yet, I dare venture to affirm, that I have not omitted the fubftance of any Propofition which will ever be referred to, by Authors in any other Science.

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