10-17-39 Hem. PREFACE. OS authors, from a natural to render their fub jects as compleat as poffible, are in danger of being betrayed into prolixity: An attention to minute circumstances may be neceffary in fome kinds of compofition, but prolixity is altogether inexcufable in a scientific writer. His object is to explain the principles of science in the moft fimple and perfpicuous manner. To accomplish this end, every fuperfluity of language and reasoning ought to be ftrictly guarded againft. Whoever has attended to books of fcience will readily allow, that most of them are capable of abridgement; and that this abridgement, instead of obfcuring, or rendering the fubject more difficult, will make it more clear and intelligible to the generality of stu dents. Simplicity and concifenefs are peculiarly neceffary in communicating the Elements of fcience, which are always lefs interefting to the ftudent than the practical parts. If the author be tedious in this article, the mind, being entirely unacquainted with the utility or application of elementary truths, is apt to revolt and abandon the ftudy. But fimplicity and concifeness are more indifpenfible in the elements of mathematics than any other science. Unfortunately, however, too little attention has hitherto been given to this circumstance. Euclid, an author long and juftly admired for the excellency of his general method, has often gone fo minutely to work in his demonftrations, as to render many plain propofitions not only tedious, but difficult. His manner of demonftrating is unqueftionably the beft that has yet appeared, and therefore ought to be followed: But it is by no means impoffible to make his demonstrations as plain in much fewer words, and even to arrange many of them in a different manner, without doing the least injury to his principles. This tafk I have undertaken in the following sheets. If I have fucceeded, one capital objection to the fudy of mathematics is happily removed, as the Elements of Euclid may now be learned in one half of the ufual time, and with greater eale to the student. That the reader may be the better prepared for the alterations he may meet with, I have here mentioned a few, with the reafons which induced me to make them. Book : Book I. ax. 10. "Two right lines do not bound a figure;" in ftead of "include a fpace," the boundaries of space, being difputed by metaphyfical writers, become unfit for a mathematical axiom. Prop. 5. which is rather too tedious, I have proved from prop. 4. in very few words, and have not used more freedom than is done in the demonftration as it now ftands. The fecond part, viz. the angles below the bafe, I have left out till the 13th is proved, from which it eafily follows; and likewife in proving the bafes equal in the 4th, I have changed the indirect proof, and given a direct one, by which it is both fhorter and easier comprehended. The manner in which have enunced_the_7th prop. renders the fecond part of the 5th unneceffary; yet have fuppofed no more given than what must be fuppofed before a proof can be begun. But, those who think it ought to be in more general terms, I have indulged in the 21ft, from which it naturally follows. As fome have thought axiom 12. not selfevident, and therefore ought not to be an axiom, I have added a cor. to prop. 17. that convincingly proves it. The 35th and 37th are joined in one, as nothing can follow more naturally than, if the wholes are equal, their halfs are likewife fo. The fame may be faid of the 36th and 38th; nor is it less natural to prove it from half the parallelogram than to double the triangle, and then take its half. I cannot agree with Mr Simpson in leaving out the corollaries from prop. 32. nor can I find any reafon for his fo doing. Book II. I have varied the enunciation of several of the propofitions, and expreffed them in clearer terms. In the 8th propofition, the equality of the fquares is proved in a fhorter but clearer manner than that prefently ufed. The 13th is retained much in the fame manner as in Commandine's Euclid; for, though it be true of every side of a triangle fubtending an acute angle; yet, as the demonftration is general, and the perpendi cular falling within or without the triangle, makes no real alteration, proving it in different figures becomes unneceffary. Book III. The firft definition is challenged by Mr Simpfon, which, he says, ought to be proved; for this I can see no reason, or any neceffity of a proof, as the equality of coincident figures is admitted, ax. 8. Book I. I have taken another demonftration in place of that used in the 2d propofition, which I thought as mathematical as that used either by Commandine or Simpfon, and much shorter. To the 8th prop. I have added, "that only two << equal lines can fall either the convex or concave part of "the circumference ;" but the demonftration of the whole is fhorter than that prefently ufed. In the 16th, "the angle of a "femicircle" is omitted, because it follows more naturally as a corollary. The 18th and 19th are joined in one, for the reafons already given. I have put a fhort and natural demonftra upon tion in place of the 2d part of prop. 21. and changed the figure. The 25th is fhortened, and the 28th and 29th joined in one. In the 31ft," the angle of a fegment" is left out, but resumed in the cor. as it follows naturally from the propofition. I have added a cor. to prop 37. which is found neceflary in practice. Book IV. is much fhortened, the 12th, 13th, and 15th, are demonftrated in a different manner. Book V. is shortened almost in every propofition. In Book VI. I have added a few words to the 5th def. which renders it compleat; the lemma added to prop. 22. is therefore unneceffary; as alfo def. A. inferted after def. 11. book V. by Mr Simpson. The 5th and 6th propofitions are joined in one, as alfo the 14th and 15th; the demonftrations are in ge neral shorter. Book XI. Def. 10. is retained, as univerfally true, for the reafons given in the note at the end of the preface. Prop. 7. As this propofition has no dependence on any of the preceding propofitions of this book, I have put it in place of the 6th, and joined the 6th and 8th in one, by which the proposition is made both fhorter and plainer than when feparate. The greatest part of the propofitions of this book are confiderably shortened. Book XII. Prop. 5. and 6. are joined in one, and much shortened, and the demonftrations in part new. The 8th and 9th are demonftrated in a much fhorter and more familiar manner; the greatest part of the 10th and 11th being only a repetition of the 2d, that Prop. is only referred to, as it is not neceffary to demonftrate a prop. twice over, nor has Euclid done fo any where at fo great length as in this book. In PLAIN TRIGONOMETRY I have not inferted any thing that depends for illuftration on infinite feries, that being a fubject more proper for the higher parts of mathematics; but have rendered the elements fhort and comprehenfive, so as fully to contain the principles of trigonometry, as well as to explain the nature and use of the logarithmic canon. In SPHERICAL TRIGONOMETRY, the propofitions are demonstrated in a fhort and eafy method, from the principles of plain trigonometry. The obfervations made on them by Mr Cunn are left out, being wholly contained in the propofitions, and what he intends by them eafily difcovered in practice. I have added a fhort explanation, of the nature and use of Sines, Tangents, Secants, and verfed Sines, both natural and artificial; and how to change Briggs's Logarithms to the Hyperbolic, and vice verfa, with examples of the above. To which are annexed TABLES of the Logarithms of Numbers, of Sines, Tangents, and Secants, both natural and artificial, which will work to the fame exactnefs, of any extant, even to fecond and third minutes, or farther, if thought necessary. Upon Upon the whole, although the above alterations are intended to render the elements eafier and fooner acquired, yet are not intended to indulge the indolence of either mafter or ftu. dent. The Elements of Geometry being of fuch extensive use, that a thorough knowledge of them is abfolutely neceffary, whether in the literary or mechanic profeffion; the concifenefs of the reafoning, and conclufiveness of the arguments, render that knowledge a neceffary qualification for the pulpit or bar; and in profecuting the fciences, this knowledge becomes abfolutely neceffary but the fooner it can be acquired, a thorough knowledge of it may more eafily be attained: and what is referved of that time, which even an experienced Teacher would formerly have taken up in barely demonftrating the propofitions, may be employed in pointing out their particular beauties, the accuracy of the reasoning, their ufe in the affairs of life, and their application to the fciences, which will be of great advantage to the ftudent, as he is hereby let into the beauties of the science by the time he formerly could have had but even a tolerable knowledge of the method of demonftration. The author does not hereby mean to infinuate, that this work is without exception; that notwithstanding the pains he has taken to render it as correct as poffible, yet feveral inaccuracies, both in the language and demonftrations, may have escaped his notice, which he hopes the learned will excufe, and lend their affiftance to render it more useful, if they fhall think it worthy of another impreffion. That Mr Simpfon has fallen into a mistake, in the demonftration he has given to prove the falfity of def. 10. Book XI. will appear from the following obfervations: He has proved that the triangles EAB, EBC, ECA, contain ing the one folid, are equal and fimilar to the three triangles FAB, FBC, FCA, containing the other folid, and having the common bafe ABC; he does not deny the equality of thefe fo lids, but compares them with another folid contained by three triangles GAB, GBC, GCA, and common bafe ABC, which three triangles he neither proves equal nor fimilar; but concludes, that the folid contained by the three triangles GAB, GBC, GCA, is not equal to the folid contained by the three triangles EAB, EBC, ECA, and common bafe ABC, because the one contains the other. If he had proved, that the triangles GAB, GBC, GCA, were equal and fimilar to the other three triangles EAB, EBC, ECA, and common base ABC, and then proved the folids not equal, he would then have gained his point; but as he has not even so much as attempted this, def. 1o. must be held as univerfally true; at least till fome better argument is produced against it. But as he fuppofes it proved not univerfally true, he prefents us with prop. A, B, C, after prop. 23. Book XI. to supply its defect. prop. C. "Solid figures contained by the fame num"ber of equal and fimilar planes alike fituated, and having none "of their folid angles contained by more than three plane angles, "are equal and fimilar to one another." But this prop. C. will evidently appear infufficient to supply this fuppofed defect, on account of the limited fenfe in which it is taken; for, if solid figures, bounded by an equal number of equal and fimilar planes, are not equal and fimilar, but under this limitation, then prop. 15. Book V. muft not be univerfally true, which I fuppofe will not easily be admitted; and, if not admitted, then prop. C muft be a very infufficient foundation for proof of the following propofitions depending on it, viz. Prop. 25. 26. and 28. and confequently eight others, viz. 27th, 31ft, 32d, 33d, 34th, 36th, 37th, and 40th. Book XI. all which are by this author toffed off their bafe, which is univerfally true, and placed upon this limited one. Mr Simpfon farther objects, that though this definition be true, yet ought not to be a definition, but a proposition, and the truth of it proved. The fame objection might be made with equal propriety to feveral others; for example, why not prove the equality of thefe angles which determine the equal inclination of planes, Def. 7. Book XI. and the equality of right lines cqually diftant from the center, both which we may conclude to be Euclid's, as Mr Simpfon does not object to them; for he would make us believe none are Euclid's that he does not affirm to be so, and that frequently without any other reafon given for it, but his own ipfe dixit. If we confider the nature of a definition, it is, if I miftake not, diftinguishing bodies from one another, by fuch properties as cannot be applied to any other bodies, but thofe it is intended to diftinguish. In which fenfe, if the properties given in this definition are fuch as diftinguish fimilar and equal bodies from others that are not fo in every inftance, then it is cer tainly a proper definition; but Euclid has fometimes thought proper to prove his definitions; for example, def. 4. Book III. which he has proved, prop. 14. of that book. This, it would appear, he has not thought neceffary to prove, probably, if we may be allowed to affign a reason in his name, that he has thought it so self-evident, that none would ever call the truth of it in queftion; but as the truth of it has been called in question, the definition may be proved in the following manner from Mr Simpfon's demonstration to prove the contrary; for which obferve his own figure and demonstration. He has proved the three triangles EAB, EBC, ECA, containing the one folid, equal and fimilar to the three triangles FAB, FBC, FCA, containing the other fo lid |