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25th is again introduced the same general properties of Frie angles, which are evidently deducible from the fourth. For, having, in the fourth, proved, that, in two Triangles, if there are found two Sides, in each, respectively equal, and that the Angles, contained between the equal Sides, are also equal, the remaining Sides are equal; can any Person of common sense, who knows what a Plane Angle means, want Demonstration, that if, in the Case of equality of the Angles, the sides opposite those Angles are equal; if one of the Angles be greater or less, the side opposite will also be greater or less? and, vice versa (in the 25th) if the Side or Base, lying between the equal Sides, be greater or less, the Angle it subtends must necessarily be so too: these things seem, to me, such a necessary consequence, that I would not hesitate to take them for granted.
This Book is much encumbered with useless Demonstrations, of feveral Propositions, already fully demonstrated or need none; for, what is evident in itself, needs no other Demonstration; and such Propositions, as may easily and clearly be deduced from some preceding ones, are as well made Corollaries to them; such are all converse Propofitions; for I cannot conceive it useful to perplex and embarrass the minds of Youths, with demonftrating what is not essential and absolutely necessary.
I have, therefore, greatly abridged, and altered the Elements of Euclid's first Book; yet, I am greatly mistaken, if I have not retained all that is essential in it; and which, I think, may be much easier and more regularly acquired. After abstracting the 14 Problems, I have but 20 other Propositions in this first Book ; three of which number (the 3d the 6th and 16th) are not in Euclid; whereas, exclusive of the Problems, Euclid has 34 Theorems, twice the number, of the remaining seventcen.
The 7th of Euclid is not only useless, in itself, but the Demonstration is intricate, to a beginner; and since the 8th, which is made to depend on it, is clearly deducible from the 4th, I see no use for it all. In respect of the 4th (my 8th) I cannot think the Demonstration of it perfectly geometrical; but, such as it is, has been generally accepted; and although it has been thought somewhat deficient, yet, I have not seen even an attempt at any other kind of Demonstration; which is fufficient conviction that no other can be given. As it is the first Proposition on the properties of Figures, which might well pass for an Axiom, the Demonftration, being purely mental, is fatisfactory; though it fupposes a manual, or mechanical application of one to the other; the proof arising, from which, would be, mere, ly, occular.
Having received the first rudiments of Geometry from Pardie, I may perhaps have imbibed a prejudice, in respect of his manner of treating the first Elements. Upon the whole, I look on that Tract as very imperfect and irregular; but I have always thought it more introductory to treat, first, of Lines and Angles, before Figures ; keeping them distinct and unconnected, as much as possible. The 13th. of Euclid I have, therefore, made the first; for, on the foundation of it, depends, in a great measure, the whole first Book. I am the more confirmed in it, having seen (fince my Plan was formed) two or three Authors, of some Fame, who have pursued the same Plan, nearly; which at first (I freely own) hurt me, not a little, but, which, I am now reconciled to, and think it does me credit.
According to Euclid, the properties of parallel Lines &c. cannot be obtained without having recourse to Triangles, and is the reason why the 27th, and the following, are not introduced sooner, which are previously necessary to demonstrate the 32nd; in which, the 16th and 17th are more fully and perfectly demonstrated: wherefore, Euclid
was obliged to have recourse to the 16th, as a Lemma, (which is much more complex and difficult to conceive than the 32nd) in demonstrating several Propositions previous to them. But, for what use the 17th is introduced, I am at a loss to devise, it being never once referred to, I believe, in the whole Elements; it might (if it be necessary) be a Corollary after the 32nd, and it is never used before. We can never be at a loss to know, that any two Angles of a Triangle are less than two Right Angles, having full Demonstration (in the 32nd) that, the three Angles of every Triangle are, together, equal to two Right ones; confequently it is of no use at all.
In the 18th of this, is contained the 35, 36, 37 and 38th of Euclid including also, in a Corollary, the 39th and 40th.
This property of Parallelograms and Triangles is, undoubtedly, very extensive; but it is all included in the first part of the 18th (i. e. in the 35th of Euclid) having previously demonstrated, in the 17th, that every Triangle is equal to half a Parallelogram, having the same Base and Altitude; which, Euclid deduces and demonstrates from the other. To dwell so long on one Property, in fix Propofie tions, becomes tedious, if not triling; for where is the difference, whether Parallelograms, or Triangles, have the fame or an equal Base?
The alterations, which I have made in the first Book, are very considerable; and the manner of demonstrating is, in geaeral, different, and more concise; as to the clearness of it, it must be left for others to determine. I do not doubt but the judicious, candid, and impartial Reader will concur with me, or not disapprove, entirely, the liberty I have taken. I thought it incumbent on me to advertise him of the alterations, and, by pointing them out, he may, with more ease, form a judgment of them.
The first fix Propositions, and the Corollaries deduced from them, contain all the properties of Right Lines, and Angles formed by their Intersections; which are effentially necessary to be known, before we can obtain a knowledge of the properties of Piane Figures. But, I think it scarce worth the loss of time to demonstrate them, they being in a manner self evident; infomuch, that the greatest part of them may be, and are, by some, given as first Principles, ör self evident Propofitions.
1. For, since (by Def. 10.) one Right Line standing perpendicularly on another makes the Angles on each side equal to the other, they are, therefore, Right Angles. The Perpendicular, CD, being com
E mon to both Angles, ACD and DCB; the two other Sides AC, CB are, con
A sequently, in one Right Line, ACB. And, it is clearly evident, that any other Line, or number of Lines, as CE; CD, CF, at the same Point C, mult
H make all the Angles, ACE+ECD+DCF + FCB, equal to the two Right Angles (ACD+DCB.)
Hence, the first Proposition and its Corollaries are evidently clear and manifest,
2. Again; (by Def. 11.) if the Perpendicular, CD, be produced, towards H, there is generated equal Angles, to the opposite, on either Side, for, they are all Right Angles, consequently equal, by the gth Axiom.
So likewise, if any other, inclined, Line be produced, as EC, or FC, it will, necessarily, produce equal Angles, to the opposite, on the other side of AB; as ACG equal FCB, and GCB=ACF; and which, together, are also equal to two Right Angles.
For, if EC bisects the Right Angle ACD, CI will also bisect the oppofite, HCB; and, if CF trisect the Right Angle DCB; the opposite Angle, ACH, will also be triseeted, if FC be produced, towards G; and so, in whatever proportion DCB is divided, by FC, the continuation of FC will, necessarily, divide the opposite, ACH, in the fame Proportion; wherefore, the Ang. GCH-DCF, and ACG=FCB; also, GCH+HCB-DCF + ACD; and ACG+GCB=FCB+ACF; i. e. they are each equal to two Right Angles; and consequently, all the Angles about the Point C are equal to four Right Anglés.
Hence, it is plain, that the first and second Propositions, with their Corollaries, are clearly deducible from the 10th and 11th Definitions ; after what has been said in the Theory of Plane Angles is well understood.
3. It is also, I prefume, easy to be conceived, and whică I think no Person, at least, who has any Talent for Geometry) can be at a loss to conceive; that, two or more parallel Lines must necessarily have the same Inclination to any Right Line which cuts them all; i. e. the Angle of Inclination, with each Line, is equal to one another; the bare Idea of Parallelism seems to indicate it; which being known, or understood, all the rest follow of course, viz. that the Alternate Angles are equal to one another, and the two internal Angles, on each Side, are equal to two Right Angles. D)
For, fuppose the Right Lines AB, CD, and EF to be parallel amongst
themselves, and all cut by any Right 2
Line, GE; making, with them, the " Angles X, Y, Z; and, fuppose AB to
be moved, directly, to CD, still keeping parallel to CD, it must necessarily coincide with CD; and, consequently,