PROP. II. Two straight lines, which make with a third line the interior angles on the same side of it less than two right angles, will meet on that side, if produced far enough. Let the straight lines AB, CD, make with AC the two angles BAC, DCA less than two right angles; AB and CD will meet if produced toward B and D. In AB take AF=AC; join CF, produce BA to H, and through C draw CE, making the angle ACE equal to the angle CAH. Because AC is equal to AF, the angles AFC, ACF are also equal (5. 1.); but the exterior angle HAC is equal to the two interior and opposite angles ACF, AFC, and therefore it is double of either of them, as of ACF. Now ACE is equal to HAC by construction, therefore ACE is double of ACF, and is bisected by the fine CF. In the same manner, if FG be taken equal to FC, and if CG be drawn, it may be shewn that CG bisects the angle ACE, and so on continually. But if from a magnitude, as the angle ACE, there be taken its half, and from the remainder FCE its half FCG, and from the remainder GCE its half, &c. a remainder will at length be found less than the given angle DCE.* Let GCE be the angle, whose half ECK is less than DCE, then a straight line CK is found, which falls between CD and CE, but nevertheless meets the line AB in K. Therefore CD, if produced, must meet AB in a point between G and K. Therefore, &c. Q. E. D. This demonstration is indirect; but this proposition, if the definition of parallels were changed, as suggested at p. 302, would not be necessary; and the proof, that lines equally inclined to any one line must be so to every line, would follow directly from the angles of a triangle being equal to two right angles. The doctrine of parallel lines would in this manner be freed from all difficulty. PROP. III. 29. 1. Euclid. If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another; the exterior equal to the interior Prop. 1, 1. Sup. The reference to this proposition involves nothing inconsistent with good reasoning, as the demonstration of it does not depend on any thing that has gone before, so that it may be introduced in any part of the Elements. and opposite on the same side; and likewise the two interior angles, on the same side equal to two right angles. Let the straight line EF fall on the parallel straight lines For if AGH be not equal to GHD, let it be greater, then adding BGH to both, the angles AGH, HGB are greater than the angles DHG, HGB. But AGH, HGB are equal to two right angles, (13.); therefore BGH, GHD are less than two right angles, and therefore the lines AB, CD will meet, by the last proposition, if produced toward B and D. But they do not meet, for they are parallel by hypothesis, and therefore the angles AGH, GHD are not unequal, that is, they are equal to one another. Now the angle AGH is equal to EGB, because these are vertical, and it has been also shewn to be equal to GHD, therefore EGB and GHD are equal. Lastly, to each of the equal angles EGB, GHD add the angle BGH, then the two EGB, BGH are equal to the two DHG, BGH. But EGB, BGH are equal to two right angles, (13. 1.), fore BGH, GHD are also equal to two right angles. Therefore, &c. Q. E. D. there The following proposition is placed here, because it is more connected with the First Book than with any other. It is useful for explaining the nature of Hadley's sextant; and though inv lved in the explanations usually given of that instrument, it has not, I believe, been hitherto considered as a distinct Geometric Proposition, though very well entitled to be so on account of its simplicity and elegance, as well as its utility. THEOREM. If an exterior angle of a triangle be bisected, and also one of the interior and opposite, the angle contained by the bisecting lines is equal to half the other interior and opposite angle of the triangle. Let the exterior angle ACD of the triangle ABC be bisected by the straight line CE, and the interior and opposite ABC by the straight line BE, the angle BEC is equal to half the angle BAC. The lines CE, BE will meet; for since the angle ACD is greater than ABC, the half of ACD is greater than the half of ABC, that is, ECD same side of BC on which the triangle ABC is. Let them meet in E. Because DCE is the exterior angle of the triangle BCE, it is equal to the two angles CBE, BEC, and therefore twice the angle DCE, that is, the angle DCA is equal to twice the angles CBE, and BEC. But twice the angles CBE is equal to the angle ABC, therefore the angle DAC is equal to the angle ABC, together with twice the angle BEC; and the same angle DCA being the exterior angle of the triangle ABC, is equal to the two angles ABC, CAB, wherefore the two angles ABC, CAB are equal to ABC and twice BEC. Therefore, taking away ABC from both, there remains the angle CAB equal to twice the angle BEC, or BEC equal to the half of BAC. Therefore, &c. Q. E. D. BOOK II. THE Demonstrations of this Book are no otherwise changed than by introducing into them some characters similar to those of Algebra, which is always of great use where the reasoning turns on the addition or subtraction of rectangles. To Euclid's demonstrations, others are sometimes added, serving to deduce the propositions from the fourth, without the assistance of a diagram. PROP. A and B. These Theorems are added on account of their great use in geometry, and their close connection with the other propositions which are the subject of this Book. Prop. A is an extension of the 9th and 10th. BOOK III. DEFINITIONS. The definition which Euclid makes the first of this Book is that of equal circles, which he defines to be "those of which the diameters "are equal." This is rejected from among the definitions, as being a Theorem, the truth of which is proved by supposing the circles applied to one another, so that their centres may coincide, for the whole of the one must then coincide with the whole of the other. The converse, viz. That circles which are equal have equal diameters, is proved in the same way. The definition of the angle of a segment is also omitted, because it does not relate to a rectilineal angle, but to one understood to be contained between a straight line and a portion of the circumference of a circle. In like manner, no notice is taken in the 16th proposition of the angle comprehended between the semicircle and the diameter, which is said by Euclid to be greater than any acute rectilineal angle. The reason for these omissions has already been assigned in the notes on the fifth definition of the first Book. PROP. XX. It has been remarked of this demonstration, that it takes for granted, that if two magnitudes be double of two others, each of each, the sum or difference of the first two is double of the sum or difference of the other two, which are two cases of the 1st and 5th of the 5th Book. The justness of this remark cannot be denied; and though the cases of the Propositions here referred to are the simplest of any, yet the truth of them ought not in strictness to be assumed without proof. The proof is easily given. Let A and B, C and D be four magnitudes, such that A=2C, and B=2D; then A+B=2.(C+D). For since A=C+C, and B=D+D, adding equals to equals, A+B= (C+D) + (C+D)=2(C+D). So also, if A be greater than B, and therefore C greater than D, since A=C+C, and BD+D, taking equals from equals A-B (C-D)+(CD), that is, A-B=2(CD). BOOK V. THE subject of proportion has been treated so differently by those who have written on elementary geometry, and the method which Euclid has followed has been so often, and so inconsiderately censured, that in these notes it will not perhaps be more necessary to account for the changes that I have made, than for those that I have not made. The changes are but few, and relate to the language, not to the essence of the demonstrations; they will be explained after some of the definitions have been particularly considered. DEF. III. The definition of ratio given here has been greatly extolled by some authors; but whatever value it may have in the eyes of a metaphysician, it has but little in those of a geometer, because nothing concern 66 ing the properties of ratios can be deduced from it. Dr. Barrow has very judiciously remarked concerning it," that Euclid had probably "no other design in making this definition, than to give a general summary idea of ratio to beginners, by premising this metaphysical defi"nition, to the more accurate definitions of ratios that are equal to one "another, or one of which is greater or less than the other: I call "it a metaphysical, for it is not properly a mathematical definition, "since nothing in mathematics depends on it, or is deduced, nor, as I "judge, can be deduced, from it." (Barrow's Lectures, Lect. 3.). Dr. Simson thinks the definition has been added by some unskilful editor; but there is no ground for that supposition, other than what arises from the definition being of no use. We may, however, well enough imagine, that a certain idea of order and method induced Euclid to give some general definition of ratio, before he used the term in the definition of equal ratios. DEF. IV. This definition is a little altered in the expression: Euclid has it, that "magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the greater." DEF. V. One of the chief obstacles to the ready understanding of the 5th Book of Euclid, is the difficulty that most people find of reconciling the idea of proportion which they have already acquired, with the account of it that is given in this definition. Our first ideas of proportion, or of proportionality, are got by trying to compare together the magnitude of external bodies; and though they be at first abundantly vague. and incorrect, they are usually rendered tolerably precise by the study of arithmetic; from which we learn to call four numbers proportionals, when they are such that the quotient which arises from dividing the first by the second, (according to the common rule for division), is the same with the quotient that arises from dividing the third by the fourth. Now, as the operation of arithmetical division is applicable as readily to any two magnitudes of the same kind, as to two numbers, the notion of proportion thus obtained may be considered as perfectly general. For, in arithmetic, after finding how often the divisor is contained in the dividend, we multiply the remainder by 10, or 100, or 1000, or any power, as it is called, of 10, and proceed to inquire how oft the divisor is contained in this new dividend; and, if there be any remainder, we go on to multiply it by 10, 100, &c. as before, and to divide the product by the original divisor, and so on, the division sometimes terminating when no remainder is left, and sometimes going on ad infinitum, in consequence of a remainder being left at each. operation. Now, this process may easily be imitated with any two Rr |