we may not as well cut off from the given Line (AB) at once, the measure of C, (at E) without first placing it from A to E B D? (See the Figure) nor can I perceive that the Demonstration is, at all, c the clearer for it. As the knowledge I have acquired, in such studies, has been entirely from Books, without any other assistance ; I may, perhaps, have been able to see deficiences or superfluities in them, better than those who have studied under a Tutor ; and, I dare venture to affirm, that very few, who are not already tolerably versed 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 Treatise, endeavoured to render every Proposition easy and intelligible, to any capacity ; and have omitted none that are useful, or necessary to demonstrate the rest; and, I will be bold to affirm, that those who cannot, from it, acquire a knowledge of all that is requisite; in Plane Geometry, will never be able to attain it at all. Where there is not a capacity to understand, they had better defist from the undertaking. An equal talent is not given to all ; and, for such as have not a talent and a natural propensity, to expect to attain any tolerable share of knowledge, in Geometry, is aiming at impossibilities. The 16th and 17th Propositions, Bock ift, are entirely useless; for since, 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 seems absurd to prove, before hand, that any two of the Angles are less than two right Angles, and, that the external Angle, is greater than either of the remote ones. As if one should 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 alier, in some measure, the Elements of the first Book; that, by a different arrangement, in transposing the places of of fome Propofitions, others, which depend on them, may with more ease and elegance be demonstrated. I shall ever be of opinion with Tacquet, and some others, that, to attempt a formal Demonstration of Propositions which are self-evident, is involving a thing, in itself clear and conspicuous, in darkness and obscurity. I have always found more difficulty in demonstrating, to another Person, self-evident Propofitions, than the most intricate of others; and, when done, have only confused the Idea which the Pupil had more perfe&tly from inspection of the Figure. If a knowledge of several properties of Figures be acquired, which is necessary to elucidate other more sublime Propositions, is not the eafieft method the most eligible ? certainly it is ; and, barely to be told several properties of Triangles is sufficient for attaining the rest. Therefore, in this Work, I have reduced some Propofitions into Corollaries; as they are but a certain consequence of the preceding Proposition, or of some other. Dr. Keill, in his Preface to his Translation of Commandine, and also Mr. Cunn seems to think it an unpardonable fault in Tacquet, to omit the Demonftrations of the 5th Book; and asserts, that not one Demonstration, 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, myself, a living witness against such his affertion ; for, I do aver, that, without any other de. monstration 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 Treatise bear witness. I never could, at first, have had patience to go through the dry and tedious Demonstrations, delivered in 25 Propositions, of the 5th Book of his Euclid; and, had it first fallen in my way, I should certainly have lain it aside 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 absolutely necessary, in order to obtain a competent knowledge in Geometry, and of Proportion in particular, to tread exactly in the same steps with Euclid; therefore, I have made made free to deviate, where I think it for the better, a readier My reason for separating Practical Geometry from the Theorems pre- To make the Work more compleat. I have added, after Practi- cal Geometry, a brief Theory of the nature and construction of an Ellipfis, that most useful and valuable Figure. And also, as an Appendix to the whole, I have given a concise Theory of mensuration of Superficies and Solids; Thewing their immediate and absolute dependence on Geometry. I have for some time debated, with myself, whether I should publish a tract of Geometry or not; from which I have been de- terred, through the persuasions of some particular Acquaintance ; alledging, that there are more learned Treatises already published, than any I could produce : the truth of which, I readily assent to; but, I don't find that they are more intelligible for that, and have, therefore, determined to publish. My chief reason, for which, is to bring the whole Elements into less compass, and to abridge it of that tedious prolixity, which is in many Authors, on that Subject; in dwelling too long on Puch Propositions as are clear and evident of themselves; and, by that means, to render the study of Geome- try more pleasant and entertaining. 'Tis enough to deter any one from the pursuit of a Science, who find, in the Rudiments of it, so so much perplexity; for, I am persuaded, that, unic is a Person has a strong inclination to it, and a good natural Capacity, 'tis not a very pleasing Study, at the first, until they begin to feel the sublime 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 Demonstration, that any two fides of a Triangle, are greater than the third ? is there a Person so 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 saw one of them traverse two sides 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 greatest Angle of every Triangle, must be opposite to the greatest Side? and can there be any need to demonstrate the converse ? that the greater Side subtends, or is opposite to the greater Angle; is not the one contained in the other ? it is trifling, to no purpose ; for, all converse Propofitions may be Corollaries to the former. If two sides of a Triangle, equal to two Sides of another Triangle, contain a greater or less Angle, the Base will be greater or less ? Are not all these, and several more, obvious and clear, from a bare inspection of the Figure? nay even without it; 'tis enough to be told they have such properties, and not to lose time in trying the patience of the Student, with a tedious and puzling Demonftration of what he saw clearer before ; for, if the thing is seen or known, what needs there more? is it intended to perplex, only, where it can be of no use? to dilgult the beginrer, before he is able to see any of the Beauties id contains ? Yet, I do not omit these entirely, because the whole Elements depend on them; but have endeavoured to treat them in as simple a manner as the nature of the Subject will admit of; if I have been concise, I doubt not I shall be excufed, if I have but said enough. But, as I think I have said enough, in this place, I shall straightway proceed to the Subject ; through which, if the Reader be in. clined so follow, with an intention to learn what it contains, I am much miftaken if he loses his labour ; and, for such as peruse it only with intent to cavil, I hope they will be greatly disappointed, and find but little to cavil at. I am not so vain as to suppose it is without defeci, or that it will please all, for that I know is impoflible; but, if I have made it intelligible and useful to common capacities, it is what I aimed at, and shall rest satisfied in the fupposition that my labour is not entirely loft. much The greatest fault in Tacquet is, that his Figures are too sma!! and trivial; and it is a general fault, that they are often incorrect and frequently contradict the description. Correct and well adapted Schemes are certainly of some consequence, in which, I have been very particular ; and have, allo, carefully revised the letter press, so that, I hope there are but few errors have escaped my observation. For, Errors, in misplacing the Referrences to the Schemes, and sometimes 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 occasions ; especially, when the subject is quite new to the Student. But, if any should 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 Treatise, deviated greatly from Euclid, in many particulars, I have endeavoured to make it generally useful; by putting his Numbers after mine, to each Proposition, and also by means of an Index, I have shewn where to find any Proposition of Euclid; which, in case of reference, to Euclid, in other Works, may be readily turned to. I have well considered and digested every Proposition, have carefully revised them over and over with the strictest attention, and I am fully convinced, that there is no omission of any thing that is essential, or necessary 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 substance of any Propofition · which will ever be referred to, by Authors in any other Science. |