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From D, a point without the circle ABC, let the straight line DB be drawn touching the circle, and DCA cutting it: the rectangle AD.DC is equal to the square of DB.
Join BĀ, BC. Then, in the triangles DBA, DCB, the angle D is common, and (III. 32.) the angle A is equal to CBD; wherefore (VI. 4. cor.) AD: DB:: DB : DC; and therefore (VI. 17.) AD.DC = DB2.*
ELEMENTS, BOOK IV.
This may be demonstrated in the following easy manner by means of proportion, the circle being unnecessary.
By construction, AB.BC = AC = BD2 ; whence (VI. 17.) AB :BD::BD: BC; and therefore (VI. 6.) the angle BCD is equal to ADB, or its equal B. Hence (I. 6.) CD is equal to BD, or to its equal AC, and (I. 5.) the angle A is equal to ADC. Now BCD, which is equal to B, being equal (I. 32.) to both these taken together, is double of one of them : therefore B is double of A.
ELEMENTS, BOOK V. SEVERAL minute changes, and, it is hoped, improvements, have been made in this book, especially in the definitions. These it is unnecessary to specify; and I shall merely subjoin some remarks,
The study of Euclid would be rendered more easy to the beginner, if, in accordance with some of the preceding Notes, he should omit in the first reading the 44th and 45th propositions of the first book, and all in the second book after the first three : then, omitting in like manner, the 35th, 36th and 37th of the third book, and perhaps all the fourth book, he may proceed to the fifth and sixth books ; and, having advanced in the latter to the 17th proposition, he may then turn back and read what he has omitted, availing himself of the facilities that will be afforded, in several instances, by means of the sixth book. He might also postpone the 47th of the first book : but Euclid's proof of it is so elegant and easy, as to render this unnecessary; and it is desirable, that he should be early acquainted with so important a proposition.
which will tend to illustrate, in a familiar manner, the nature of proportion generally, and in particular the fifth and sixth definitions of this book. Our common idea of proportion, and that which is presented in these definitions, are apparently very different; and it will be a principal object in the following observations to show their agreement.
In estimating the comparative magnitudes of numbers or other quantities, we generally employ a species of division.
Thus, we say, that one country has three times or four times the extent of another; and that the population of one city exceeds that of another in a fourfold degree; thus considering how often one of the magnitndes contains the other, and obtaining by that means the idea of ratio. If by such an examination we find that one country contains three times as many square miles as another, and that the population of the former is treble of that of the latter, we see that the ratios of the extents and populations are equal, and we say that the populations of the countries and their extents are proportional. The method, therefore, which naturally presents itself for estimating the ratio of one quantity to another, is to divide the former by the latter. Thus, the ratio of a line of 10 inches to one of 2 inches is 5, or is that of 5 to 1 ; and since, on the same principle, 15 has to 3 the same ratio, we say that 10, 2, 15 and 3 are proportionals. According, therefore, to this first and simplest notion of proportion, it appears that four magnitudes are proportional, when the quotients obtained by dividing the first by the second, and the third by the fourth, are equal. These quotients, in the great majority of actual instances, are fractional : still, however, if they be equal, we regard the magnitudes as proportional. Thus, 15, 6, 20, and 8 are proportional, because 15 is equal to twice 6 and the half of 6, and 20 to twice 8 and the half of 8; or, which is the same, the quotients are each 2. In like manner, 6 is to 8, as 9 to 12, each of the quotients being in this instance the fraction
Now, by the nature of division, if the second and fourth terms be multiplied by the quotient, found as above, the products will be the first and third terms: and it evidently follows, that if the first and third terms be multiplied by any number whatever, and the second and fourth by the product of that number and the same quotient, the products of the first and second terms will be equal, as will also those of the third and fourth. Thus, resuming the numbers 15, 6, 20, and 8, if we multiply 6 and 8 by 21, the products are 15 and 20; that is, 15 = 24 X 6, and 20 = 23 x 8. If we now multiply these equals by any number whatever, the products will be equal. Let us, therefore, multiply them by the denominator 2, and we obtain 2.15 = 5.6, and 2.20 = 5.8. From this it appears, that when the antecedents are multiplied by 2, and the consequents by 5, the multiples of the first and second are
equal, as are also those of the third and fourth ; a result which agrees with one of the conditions of Euclid's fifth definition : and it is plain, that if, instead of the multiples 2 and 5, we should employ their doubles, trebles, or any other numbers in the same ratio, we should still arrive at a similar conclusion. It is also plain, that if, retaining 5 as one of the multipliers, we were to use along with it any number greater than 2, the multiples of the first and third would exceed those of the second and fourth : while if 2 were retained, and with it any multiplier greater than 5 were used, the multiples of the first and third would then be less than those of the other terms : and thus we have the agreement with the two remaining conditions of Euclid's definition established in the instance which we have been considering,
In this example, 3 is common measure of the first and second terms, and 4 of the third and fourth; and the numbers 5 and 2 are obtained by dividing the terms of the two ratios respectively by these : and in the same manner, in every case in which the terms of the ratios are commensurable, we may find numbers by which, if they be multiplied, the products will be equal. If, however, these terms be incommensurable, that is, if they have no common measure, no numbers can be found which will give multiples exactly equal, as the quotient of two incommensurable quantities cannot be accurately expressed by any number either whole or fractional. We may find numbers, however, which will render the multiples as nearly equal as we please. Thus, suppose the terms of the ratio to be V2 and 1, the former being the diagonal of a square of which the latter is the side. Then, since, by extracting the square root of 2 numerically, we find 1.414.... for the approximate value of the first term, the quotient obtained by dividing the first by the second is greater than 18 and less than 15: and if we multiply the first term by 10, and the second by 14, the first of the two results is greater than the second; while, if 10 and 15 be employed as multipliers, the first product is less than the second. "If, again, 100 and 141 were employed as multipliers, and also 100 and 142, the multiples would still have the same relation, but would approach in a greater degree to the ratio of equality: and thus, by taking other multipliers, we might approximate to exact equality, as nearly as we please. Now, it is evident, that every thing would be precisely similar with respect to 2 V2 and 2, 3 V2 and 3, and so on; so that V2: 1::2V2:2::3 v2 :3, &c. It thus appears that not only in commensurable magnitudes, but also in incommensurable ones, if the quotients obtained by dividing the antecedents by the consequents be equal, the magnitudes are proportional, according to the criterion of Euclid : for if multiples be taken, as in the 5th definition, those of the antecedents will be either both greater, or both less, than those of the consequents; and if the multiples of
the first and second terms be nearly in the ratio of equality, so will those of the third and fourth.
The illustration here given is the same in substance as proposition I., page 127. Elrington, Leslie, and others, instead of Euclid's definition, bave adopted one to the following effect: The first of four magnitudes is said to have to the second the same ratio which the third has to the fourth, when any submultiple whatever of the first is contained in the second, as often as a like submultiple of the third is contained in the fourth. This definition readily arises from the supposition, that when four magnitudes are proportional, the first term is contained in the second as often as the third is contained in the fourth, which, with the exception of the order in which the terms are taken, is the same as the popular idea of proportion already considered. If, in applying this principle, we either find that the first and third terms are greater than the others, or are not contained in them without remainder, we use, instead of the first and third, submultiples of them, and we regard the magnitudes as proportionals, when the first, or any submultiple of it, is contained in the second, as often as the third, or a like submultiple of it, is contained in the fourth. This method is very simple in its application, and, with proper management, it serves to establish the theory of proportion on correct principles.
With regard to the Supplement to the fifth book, it may be remarked, that while the properties established in it are primarily those of numbers, they belong equally to lines, surfaces, and other magnitudes; as all magnitudes whatever may be conceived to be divided into parts, each equal to a common unit, and may thus be expressed by means of numbers.
ELEMENTS, BOOK VI.
INSTEAD of this definition, the following is sometimes adopted : “ Three or four straight lines are said to be in harmonical proportion, when the first is to the last, as the difference of the first two to the difference of the last two." Thus, in the figure for proposition A., page 141, the three lines DB, DG, DC are in harmonical proportion, if DB : DC::BG (= DB — DG): GC (= DG DC); which agrees exactly with the definition given in the text, if the straight line DB be regarded as divided into the three parts DC, CG, GB.
This relation of lines receives its denomination from the fact, that three musical strings of the same thickness and tension, and proportional to 1, ş, and į, produce the sounds of a certain note, its fifth, and its octave; and these lengths are such that the first is to the third, as the difference of the first and second to the difference of the second and third. By the insertion of other harmonical means between these, the lengths of the strings, giving the remaining notes of the octave, will be obtained, the lengths for the eight notes being proportional to the numbers 1, $, , 1, }, }, 15, 3; of which is an harmonical mean between 1 and f, g between 1 and , and the second of two harmonical means between 1 and f: while } is the first of two such means between and ), and it is the harmonical mean between 1 and the barmonical mean between it and f.*
PROP. XXVIII. AND XXIX.
These propositions may be omitted by the student, as they are of scarcely any use, except the cases of them which are the 35th and 36th propositions of the Appendix, Book I. From the latter propositions indeed, it is easy to derive solutions of the former. Thus, to construct the 28th, make (I. 45.) a rectangle equal to the given figure K: then to the base EC, to the altitude of CD, and to one side of the rectangle equal to K, find a fourth proportional, and complete the rectangle contained by this fourth proportional and the other side of the rectangle equal to K: lastly, divide (App. I. 35.) AB in X, so that the rectangle AX.XB may be equal to the latter rectangle, and the thing is done. The proof is easy, and may serve as an exercise to the student. The 29th will be constructed in exactly the same manner, except that the 36th, and not the 35th, of the Appendix, Book I., is to be employed.
From this proposition, and from the scholium to the eleventh of the second book, it is plain that the segments of a line cut in extreme and mean ratio, are incommensurable to one another.
The reciprocals of three equidifferent numbers, or, as they are commonly called, numbers in arithmetical progression, are in harmonical proportion. Thus, 1, lf and 2 have equal differences ; and their reciprocals, that is, the quotients obtained by dividing unity by them, are 1, f and 7, the harmonicals mentioned above. Again, if there be three harmonicals, and if the first and second be both multiplied, or both divided, by any number, and the second and third be both multiplied, or both divided, by any other number, the four numbers thus obtained are in harmonical proportion. These properties are most easily proved by means of algebra.