Book III.

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 each other, so that their centres may coincide, for the whole of one must then coincide with the whole of the other. The converse, 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 first and fifth of the fifth book. The justness of this remark cannot be denied; and though the cases of the propositions here referred to are the simplest of all, yet the truth of them ought not in strictness to be assumed without proof. 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, A+B=(C+D)+(C+D)=2(C+D). If A be greater than B, and therefore C greater than D'; then, since A=C+C, and B-D+D, A-B (CD) + (CD)= 2(C-D).

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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 which I have made, than for those which I have not made. The changes are 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 concerning 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 "definition 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 de(( pends 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, suppose 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.

Book V.

Book V.

DEF. IV.

This definition is a little altered in the expression. Euclid has it, that "magnitudes are said to have a ratio to one an"other, 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 are at first 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 which arises from dividing the third by the fourth.

Now, since 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 often 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 magnitudes A and B, provided they be of the same kind, or such that one can be multiplied so as to exceed the other. For, suppose that B is less than A; take B out of A as often as it can be found, and let the quotient be noted, and also the remainder, if there be any; multiply this remainder by 10, or 100, &c. so as to ex

ceed B, and let B be taken out of the quantity produced by Book V. this multiplication as often as it can be found; let the quotient be noted, and also the remainder, if there be any. Proceed with this remainder as before, and so on continually. It is evident that we have an operation applicable to all magnitudes whatever, and which may be performed with respect to any two lines, any two plane figures, or any two solids, &c.

Now, when we have two magnitudes and two others, and find that the first divided by the second, according to this method, gives the very same series of quotients that the third does when divided by the fourth, we say of these magnitudes, as we did of the numbers above described, that the first is to the second as the third is to the fourth. There are only two more circumstances necessary to be considered, in order to bring us precisely to Euclid's definition.

First, it is known from arithmetic that the multiplication of the successive remainders each of them by 10, is equivalent to multiplying the quantity to be divided by the product of all those tens; so that multiplying, for instance, the first remainder by 10, the second by 10, and the third by 10, is the same thing, with respect to the quotient, as if the quantity to be divided had been at first multiplied by 1000; and therefore our standard of the proportionality of numbers may be expressed thus: If the first multiplied any number of times by 10, and then divided by the second, give the same quotient as when the third is multiplied as often by 10, and then divided by the fourth, the four magnitudes are proportionals.

Again, it is evident that there is no necessity in these multiplications for confining ourselves to 10, or the powers of 10, and that we do so, in arithmetic, only for the conveniency of the decimal notation; we may therefore use any multipliers whatever, provided we use the same in both cases, Hence we have this definition of proportionals: When there are four magnitudes, and any multiple whatever of the first divided by the second, gives the same quotient with the like multiple of the third divided by the fourth, the four magnitudes are proportionals, or the first has the same ratio to the second that the third has to the fourth.

We are now arrived very nearly at Euclid's definition; for, let A, B, C, D be four proportionals, according to the definition just given, and m any number; and let the multiple of A by m, that is mA, be divided by B; and, first, let the quotient be the number n exactly; then also, when mC is divided

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nB, mC

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Book V. by D, the quotient will be n exactly.. But, when mA divided by B gives n for the quotient, mAnB by the nature of division, so that when mA =nD, which is one of the conditions of Euclid's definition. Again, when mA is divided by B, let the division not be exactly performed, but let n be a whole number less than the exact quotient, then nB < mA, or mA > nB; and, for the same reason, mC > nD, which is another of the conditions of Euclid's definition.

Lastly, when mA is divided by B, let n be a whole number greater than the exact quotient, then mA < nB; and because n is also greater than the quotient of mC divided by D (which is the same with the other quotient), mC < nD.

Therefore, uniting all these three conditions, we call A, B, C, D, proportionals, when they are such that, if mA > nB, mC > nD; if mA = nB, mĊ = nD; and if mA < nB, mC <nD, m and n being any numbers whatever. Now, this is exactly the criterion of proportionality established by Euclid in the 5th definition, and is derived here by generalising the common and most familiar idea of proportion.

It appears from this, that the condition of mA containing B, either with or without a remainder, as often as mC contains D, with or without a remainder, and of this being the case whatever value be assigned to the number m, includes in it all the three conditions which are mentioned in Euclid's definition; and hence the definition may be expressed a little more simply by saying, four magnitudes are proportionals, when any multiple of the first contains the second (with or without a remainder), as often as the same multiple of the third contains the fourth. But, though this definition is certainly, in the expression, more simple than Euclid's, it is not, as will be found on trial, so easily applied to the purpose of demonstration. The three conditions which Euclid brings together in his definition, though they somewhat embarrass the expression of it, have the advantage of rendering the demonstrations more simple than they would otherwise be, by avoiding all discussion about the magnitude of the remainder left, after B is taken out of mA as often as it can be found. All the attempts, indeed, that have been made to demonstrate the properties of proportionals rigorously, by means of other definitions, only serve to evince the excellence of the method followed by the Greek geometer, and his singular address in the application of it.