M manner. OST authors, from a natural anxiety to render their sub jects as compleat as possible, are in danger of being betrayed into prolixity: An attention to minute circumstances may be necessary in some kinds of composition, but prolixity is altogether inexcusable in a scientific writer. His object is to explain the principles of science in the most simple and perspicuous To accomplish this end, every fuperfluity of language and reasoning ought to be strictly guarded againit. Whoever has attended to books of science will readily allow, that most of them are capable of abridgement ; and that this abridgement, instead of obscuring, or rendering the subject more difficult, will make it more clear and intelligible to the generality of ftudents. Simplicity and conciseness are peculiarly necessary in communicating the Elements of science, which are always less interesting to the student 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 study. But simplicity and conciseness are more indispensible 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 demonstrations, as to render many plain propofitions not only tedious, but difficult. His manner of demonstrating is unquestionably the best that has yet appeared, and therefore ought to be followed : But it is by no means impoflible 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 task 1 have undertaken in the following sheets. If I have succeeded, one capital objection to the ftudy of mathematics is happily removed, as the Elements of Euclid may now be learned in one half of the usual 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 prop. 4. in Book I. ax. 10.“ Two right lines do not bound a figure;" instead of " include a space," the boundaries of space, being difputed by metaphysical writers, become unfit for a mathematical axiom. Prop. 5. which is rather too tedious, I have proved from very few words, and have not used more freedom than is done in the demonftration as it now stands. The second parts viz. the angles below the base, I have left out till the 13th is proved, from which it eafily follows; and likewise in proving the bases equal in the 4th, I have changed the indirect proof, and given a direct one, by which it is both shorter and easier comprehended. The manner in which : have enunced the 7th prop. renders the second part of the 5th unnecessary; yet have supposed no more given than what must be supposed before a proof can be begun. But, those who think it ought to be in more general terms, I have indulged in the 21st, 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 likewise 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 reason for his so doing. Book II. I have varied the enunciation of several of the propositions, and expressed them in clearer terms. In the 8th proposition, the equality of the squares is proved in a shorter but clearer manner than that prefently used. The 13th is retained much in the same manner as in Commandine's Euclid; for, though it be true of every lide of a triangle fubtending an acute angle; yet, as the demonstration is general, and the perpendicular falling within or without the triangle, makes no real alteration, proving it in different figures becomes unnecessary. Book III. The first definition is challenged by Mr Simpson, which, he says, ought to be proved ; for this I can see no reason, or any necessity of a proof, as the equality of coincident figures is admitted, ax. 8. Book I. I have taken another demonstration in place of that used in the 2d proposition, 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 upon the convex or concave part of “ the circumference ;" but the demonstration of the whole is shorter than that presently used. In the 16th, “ the angle of a “ semicircle” is omitted, because it follows more naturally as corollary. The 18th and 19th are joined in one, for the reafons already given. I have put a short and natural demonstra a tion in place of the ad part of prop. 21. and changed the figure. The 25th is shortened, and the 28th and 29th joined in one. In the 31st,“ the angle of a segment" is left out, but resumed in the cor. as it follows naturally from the proposition. I have added a cor. to prop 37. which is found neceffary in practice. Book IV. is much fhortened, the 12th, 13th, and 15th, are demonstrated in a different manner. Book V. is shortened almost in every propofition. In Book VI. I have added a few words to the sth def. which renders it compleat ; the lemma added to prop. 22. is therefore unnecessary; as also def. A. inserted after def. 11. book V. by Mr Simpson. The 5th and 6th propositions are joined in one, as also the 14th and 15th; the demonstrations are in general shorter. Book XI. Def. 10. is retained, as universally true, for the reasons given in the note at the end of the preface. Prop. 7. As this proposition has no dependence on any of the preceding propositions 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 shorter and plainer than when separate. The greatest part of the propositions of this book are considerably shortened. Book XII. Prop. 5. and 6. are joined in one, and much fhor tened, and the demonstrations in part new. The 8th and oth are demonstrated in a much shorter and more familiar manner ; the greatest part of the oth and 11th being only a repetition of the 2d, that Prop. is only referred to, as it is not necessary to demonstrate a prop. twice over, nor has Euclid done so any where at so great length as in this book. In PLAIN TRIGONOMETRY I have not inserted any thing that depends for illustration on infinite series, that being a subject more proper for the higher parts of mathematics; but have rendered the elements short and comprehenlive, 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 short and easy method, from the principles of plain trigonometry. The observations made on them by Mr Cunn are left out, being wholly contained in the propositions, and what he intends by them easily discovered in practice. I have added a short explanation, of the nature and use of Sines, Tangents, Secants, and versed Sines, both natural and artificial ; and how to change Briggs's Logarithms to the Hyperbolic, and vice versa, 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 same exactness, of any extant, even to second and third minutes, or farther, if thought necessary. Upon Upon the whole, although the above alterations are intended to render the elements easier and sooner acquired, yet are not intended to indulge the indolence of either master or ftu. dent. The Elements of Geometry being of such extensive use, that a thorough knowledge of them is absolutely necessary, whether in the literary or mechanic profession; the conciseness of the reasoning, and conclusiveness of the arguments, render that knowledge a neceffary qualification for the pulpit or bar; and in prosecuting the sciences, this knowledge becomes absolutely neceffary: but the fooner it can be acquired, a thorough knowledge of it may more easily be attained : and what is reserved of that time, which even an experienced Teacher would formerly have taken up in barely demonstrating the propositions, may be employed in pointing out their particular beauties, the accuracy of the reasoning, their use in the affairs of life, and their application to the sciences, which will be of great advantage to the student, 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 demonstration. 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 possible, yet several inaccuracies, both in the language and demonstrations, may have escaped his notice, which he hopes the learned will excufe, and lend their aflistance to render it more useful, if they shall think it worthy of another impression. That Mr Simpson has fallen into a mistake, in the demonftration he has given to prove the falsity of def. 10. Book XI. will appear from the following obfervations : He has proved that the triangles EAB, EBC, ECA, contain ing the one solid, are equal and similar to the three triangles FAB, FBC, FCA, containing the other folid, and having the common bale ABC ; he does not deny the equality of these solids, but compares them with another folid contained by three triangles GAB, GBC, GCA, and common base ABC, which three triangles he neither proves equal nor similar ; 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 base 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 solids 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 universally true; at least till some better argument is produced against it. But as he supposes it proved not universally true, he presents us with prop. A, B, C, after prop. 23. Book XI. to supply its defect. prop. C. “ Solid figures contained by the same num“ ber of equal and similar planes alike situated, and having nonc 6c of their folid angles contained by more than three plane angles, are equal and similar to one another.” But this prop. C. will evidently appear insufficient to supply this supposed defect, on account of the limited sense in which it is taken ; for, if solid figures, bounded by an equal number of equal and similar planes, are not equal and similar, but under this limitation, then prop. 15. Book V. must not be universally true, which I suppose will not easily be admitted ; and, if not admitted, then prop. C must be a very insufficient foundation for proof of the following propositions depending on it, viz. Prop. 25. 26. and 28. and consequently eight others, viz. 27th, 3ift, 320, 33d, 34th, 36th, 37th, and 40th. Book XI. all which are by this author toffed off their base, which is universally true, and placed upon this limit ed one. Mr Simpson 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 same objection might be made with equal propriety to several others; for example, why not prove the equality of these angles which determine the equal inclination of planes, Def. 7. Book XI. and the equality of right lines equally distant from the center, both which we may conclude to be Euclid's, as Mr Simpson 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 reason given for it, but his own ipfe dixit. If we consider the nature of a definition, it is, if I mistake not, distinguishing bodies from one another, by such properties as cannot be applied to any other bodies, but those it is intended to distinguish. In which sense, if the properties given in this definition are such as distinguish similar and equal bodies from others that are not so in every instance, then it is cere tainly a proper definition; but Euclid has sometimes 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 necessary to prove, probably, if we may be allowed to assign a reason in his name, that he has thought it so self-evident, that none would ever call the truth of it in question ; but as the truth of it has been called in question, the definition may be proved in the following manner from Mr Simpson's demonstration to prove the contrary; for which observe 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 |