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Hedger 2-5:39 37679
P R E F A C E.
OST authors, from a natural anxiety to render their sub
jects as compleat as pollible, 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 manner. To accomplish this end, every fuperfiuity of language and reasoning ought to be strictly guarded agairit. Whoever has attended to books of science will readily allow, that moit 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 itudents.
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 orn with the utility or application of elementary truths, is apt to revolt and abandon the study. But fimplicity 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 so 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 imposlible to make his de monstrations 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 I have undertaken in the following sheets. If I have succeeded, one capital objection to the fudy of mathematics is happily removed, as the Blements 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 I. ax. 10.“Two right lines do not bound a fgure;” in stead of “ include a space," the boundaries of space, being disputed by metaphysical 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 tham is done in the demonstration as it now stands. The second part, viz. the angles below the base, I have left out till the 13th is proved, from which it easily follows; and likewise in proving the bafes 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 I have enunced the 7th prop. renders the second part of the 5th unnecessary; yet have lupposed 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 some 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 so. The same may be said 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 fo doing.
Book II. I have varied the enunciation of several of the propositions, and expreffed them in clearer terms. In the Sih proposition, the equality of the squares is proved in a shorter but clearer manner than that presently used. The 13th is retained much in the same manner as in Commandine's Euclid; for, though it be true of every side of a triangle subtending an acute angle; yet, as the demonftration 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 fee no reason, or any necessity of a proof, as the equality of coincident figures is admitted, ax. 3. Book I. I have taken another demonstration in place of that used in the ad propofition, which I thouglit as mathematical as that used either by Commandine or Simpson, 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 demonitration 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 a corollary. The 18th and 19th are joined in one, for the reaLons already given. I have put a short and natural demonstra
tion in place of the 2d 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 necessary in practice.
Book IV. is much shortened, the 12th, 13th, and 15th, are demonstrated in a different manner.
Book V. is shortened almost in every proposition..
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 unnecessary; as also def. A. inserted aster 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 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 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 shortened, 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 roth and 11th being only a repetition of the 2d, that Prop. is oniy referred to, as it is not necessary to demonstrate a prop. twice over, nor has Euclid done so any where at fo great length as in this book.
In PLAIN TRIGONOMETRY I have not inserted any thiris 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 comprehensive, 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 de. monstrated 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, will 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 exactness, of any extant, even to fecond and third ininutes, or farther, if thought necedury
Upon the whole, although the above alterations are intended to render the elements ealier and tooner acquired, yet are not intended to indulge the indolence of either master or ftu. dent. The Elements of Geometry being of such extensive ufe, that a thorough knowledge of them is absolutely neceflar whether in the literary or mechanic profession; the conciseness of
; the reasoning, and conclufiveness of the argun:ents, render that knowledge a necessary qualification for the pulpit or bar; and in prosecuting the sciences, this knowledge becomes absolutely neceffary : but the sooner 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 ap. plication 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 insinuate, 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 assistance to render it more useful, if they fhall think it worthy of another impression,
That Mr Simpson has fallen into a mistake, in the demonstration he has given to prove the falfity of def. 10. Book XI. will appear from the following observations :
He has proved that the triangles EAB, EBC, ECA, containing the one folid, are equal and fimilar to the three triangles FAB, FBC, FCA, containing the other solid, and having the common base ABC; he does not deny the equality of these foJids, but compares them with another solid 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, GDC, 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 Gimilar 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. 10. must be held as universally true ; at least till some better argument is produced against it.