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“ 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.
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 AC+C. and B=D+D, taking equals from equals A-B=(C-D)+(C-D), that is, A-B=2(CD).
B00K Ꮴ, 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.
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
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 sum“mary 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.
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."
One of the chief obstacles to the ready understanding of the 5th Book of Euclid, is the difficulty that most people tind 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
magnitudes A and B, providing they be of the same kind, or such that the one can be multiplied so as to exceed the other. For, suppose that B is the least of the two ; take B out of A as oft 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 exceed B, and let B be taken out of the quantity produced by this multiplication as oft 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 ; and it is evident, that we have an operation that is applicable to all magnitudes whatsoever, and that may be performed with respect to any two lines, any two plane figures, or any two solids, &c.
Now, when we bave 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 ivided 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 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, gives 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 whatsoever, providing we use the same in both cases. Hence, we have this definition of proportionals, When there are four magnitudes, and any multiple whatsoever of the first, when divided by the second, gives the same quotient with the like multiple of the third, when 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 tour 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 by D, the quotient will be n exactly. But, when mA divided by B gives n for the quotient, mA=nB by the nature of division, so that when mA=nB, mC=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 LmA, or MA 7nB; and, for the same reason, mC7nD, 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 ZnB, and because n is also greater than the quotient of mC divided by D, (which is the same with the other quotient), therefore mcznD.
Therefore, uniting all these three conditions, we call A, B, C, D, proportionals, when they are şuch, that if mA 7 nB, mC7nD; if mA EnB, mC=nD ; and if ma 2nB, C ZnD, m and n being any numbers whatsoever. 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, whether 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 that are men. tioned in Euclid's definition
and hence, that definition may be expressed a little more simply by saying, that four magnitudes are proportionals, when any multiple of the first contains the second, (with or without remainder,) as oft 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 oft 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 than Euclid's, only serve to evince the excellence of the method followed by the Greek Geometer, and his singular address in the application of it.
The great objection to the other methods is, that if they are meant to be rigorous, they require two demonstrations to every proposition, one when the division of mA into parts equal to B can be exactly performed, the other when it cannot be exactly performed, whatever value be assigned to m, or when A and B are what is called incommensurable ; and this last case will in general be found to require an indirect demonstration, or a reductio ad absurdum.
M. D'Alembert, speaking of the doctrine of proportion, in a discourse that contains many excellent observations, but in which he has overlooked Euclid's manner of treating this subject entirely, has the following remark : “ On ne peut démontrer que de cette maniere, “ (la réduction à absurde,) la plupart des propositions qui regardent " les incommensurables. L'idée de l'infini entre au moins implicite66 ment dans la notion de ces sortes de quantités ; et comme nous n'a
“ vons qu’une idée negative de l'infini, on ne peut démontrer directe“ ment, et a priori, tout ce qui concerne l'infini mathématique." (Encyclopédie, mot Géométrie.)
This remark sets in a strong and just light the difficulty of demonstrating the propositions that regard the proportion of incommensurable magnitudes, without having recourse to the reductio ad absurdum; but it is surprising, that M. D'Alembert, a geometer no less learned than profound, should have neglected to make mention of Euclid's method, the only one in which the difficulty he states is completely overcome. It is overcome by the introduction of the idea of indefinitude, (if I may be permitted to use the word,) instead of the idea of infinity ; for m and n, the multipliers employed, are supposed to be indefinite,or to admit of all possible values, and it is by the skilful use of this condition that the necessity of indirect demonstrations is avoided. In the whole of geometry, I know not that any happier invention is to be found ; and it is worth remarking, that Euclid appears in another of his works to have availed himself of the idea of indefinitude with the same success,
viz. in his books of Porisms, which have been restored by Dr. Simson, and in which the whole analysis turned on that idea, as I have shewn at length, in the Third Volume of the Transactions of the Royal Society of Edinburgh. The investigations of those propositions were founded entirely on the principle of certain magnitudes admitting of innumerable values ; and the methods of reasoning concerning them seem to have been extremely similar to those employed in the fifth of the Elements. It is curious to remark this analogy between the different works of the same author; and to consider, that the skill, in the conduct of this very refined and ingenious artifice, acquired in treating the properties of proportionals, may have enabled Euclid to succeed so well in treating the still more difficult subject of Porisms.
Viewing in this light Euclid's manner of treating proportion, I had no desire to change any thing in the principle of his demonstrations. I have only sought to improve the language of them, by introducing a concise mode of expression, of the same nature with that which we use in arithmetic, and in algebra. Ordinary language conveys
the ideas of the different operations supposed to be performed in these de monstrations so slowly, and breaks them down into so many parts, they make not a sufficient impression on the understanding. This indeed will generally happen when the things treated of are not represented to the senses by Diagrams, as they cannot be when we reason concerning magnitude in general, as in this part of the Elements. Here we ought certainly to adopt the language of arithmetic or alge. bra, which, by its shortness, and the rapidity with which it places objects before us, makes up in the best manner possible for being merely a conventional language, and using symbols that have no resemblance to the things expressed by them. Such a language, therefore, I have endeavoured to introduce here ; and I am convinced, that if it shall be found an improvement, it is the only one of which the fifth of Eucrid will admit. In other respects I have followed Dr. Simson's edi