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(438) In a given circle (A B C D E F) to inscribe an
equilateral and equiangular hexagon.
Let G be the centre of the given circle; draw any diameter AGD; from the centre A with the radius AG describe a circle, and, from its intersections B and F with the given circle, draw the diameters B E and FC; join AB, BC, CD, DE, EF, and FA, and the figure A BCDEF is an equilateral and equiangular hexagon inscribed in the given circle.
Since the lines A B and A G are equal, as being radii of the same circle B G F, and G A and G B also equal, as being radii of the same circle ABCDEF, the triangle B G A is equilateral, and therefore the angle B G A is the third part of two right angles (XXXII, Book I.). In like manner it is proved that the triangle À G F is equilateral, and the angle A G F equal to one third part of two right angles; but the angles B G A and A G F together with FGE are equal to two right angles (XIII, Book I.), therefore F G E is one third part of two right angles, and therefore the three angles B GA, A G F, and F G E are equal, and also the angles vertically opposite to them EGD, D G C, and CGB; hence the six angles at the centre G are equal, and therefore the arcs on which they stand are equal, and the lines subtending those arcs (XXIX, Book III.); therefore the hexagon ABCDEF is equilateral, and also, since each of its angles is double the angle of an equilateral triangle, it is equiangular.
(439) It may be proved in general that every equilateral figure inscribed in a circle must be equiangular, for its angles are contained in equal arcs, and therefore stand on equal arcs. (440) The side of the regular hexagon is equal to the radius of its circumscribing circle, and its area is six times that of an equilateral triangle constructed on the radius of this circle.
If any three alternate angles A CE of the hexagon be joined by right lines, they will form the inscribed equilateral triangle.
(441) In a given circle (CAD) to inscribe an equi
lateral and equiangular quindecagon.
Let C D be the side of an equilateral triangle inscribed in the circle CAD, and CA the side of an equilateral pentagon also inscribed in the circle C A D; bisect D the arc AD; the right line joining A B is the side of 1 the inscribed quindecagon. For if the whole circumference be divided into fifteen parts, the arc CD, since it is the third part of the whole circumference, contains five of these parts ; in like manner the arc C A contains three of them, therefore the arc A D contains two, and therefore the arc A B is the fifteenth part of the whole circumference, and A B is the side of the inscribed equilateral quindecagon.
(442). I. A less magnitude is said to be a part of a greater
magnitude when the less measures the greater; that is, when the less is contained a certain
number of times exactly in the greater. The word 'part,' as applied in this definition, signifies an aliquot part or submultiple.
One quantity is said to measure another when, by continual subtraction of the former from the latter, a remainder is at length obtained equal to the former. In such a case it is plain that the former quantity multiplied by a certain integer number will become equal to the latter. Of two magnitudes thus related, the greater is said to be a multiple of the lesser, and the lesser is said to be a submultiple or aliquot part of the greater. Hence the meaning of the following definition is apparent. (443) II. A greater magnitude is said to be a multiple of a
less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly.
By the greater containing the less a certain number of times exactly, is meant, that the less is a submultiple of the greater, as already explained. (444) A greater quantity is said to contain a lesser, as often as the lesser is capable of being successively subtracted from the greater. If the greater be not a multiple of the lesser, there will be a final remainder less than the lesser quantity. The number of times the lesser is contained in the greater is expressed by that integer by which the lesser must be multiplied, in order to obtain the highest multiple of it which is contained in the greater.
The student should be cautious not to confound the expressions measures' and ' is contained in. The number 3 is ‘ contained three times in 10,' but does not measure 10, because there is a remainder 1 less than 3. Again, 3 is contained also three times in 11, but it does not measure it. On the other hand it' measures' 9, being contained in it three times exactly without any remainder.
(445) It is evident that one quantity cannot be said to be contained in or to measure another, unless they be quantities of the same kind. Thus, for example, it would be manifestly absurd to say, that a line was contained a certain number of times in a surface, linear and superficial magnitude admitting of no comparison. No increase could ren a line equal to a surface, because no increase could give it breadth, which is essential to a surface. In like manner the magnitude of a surface admits of no comparison with that of a solid, because no increase can give the one thickness which is essential to the other.
A line may be compared with a line, a surface with a surface, or a solid with a solid, as to magnitude, but none of these species can be compared with each other. This, however, does not apply to the lower species of magnitude. Different species of lines may be compared as to magnitude, because they all agree in having length only. Thus we can readily conceive a right line equal in length to a circular arc. The same applies to the different species of surfaces.
Two magnitudes A, B are said to be equimultiples of two others a, b, when a and b measure A and B respectively the same number of times. Thus the length one foot and the number 36 are equimultiples of the length one inch and the number 3 ; for an inch measures a foot twelve times, and 3 measures 36 also twelve times.
III. Ratio is a mutual relation of two magnitudes of
the same kind to one another, with respect to quantity
This definition has been by some commentators considered to be obscure and useless, and on the other hand greatly extolled by others. It is hoped, however, that the preceding observations will render it intelligible. Ratio is, in fact, the relation between two magnitudes with respect to magnitude only, that is, excluding every other property which they may have. Thus a circular arc and a straight line may agree as to magnitude, although they may differ in every other respect. When their ratio is considered, the figures, position, &c. are totally neglected, and nothing but their abstract magnitudes or lengths are considered. In the same manner we may conceive a circular arc double or triple a straight line.
The two magnitudes between which ratio subsists are stated to be of the same kind,' because if they were of different kinds,' they would not admit of any comparison as to magnitude, as has been already explained.
Two magnitudes are said to be equimultiples of two others, when they are measured by those others the same number of times.
From this definition of ratio, nothing in mathematics has been deduced. Simson thinks that it is an interpolation of some unskilful editor. We think, however, with Playfair, that finding it necessary to use the word “ ratio," Euclid thought that it was essential to that order and method for which geometry is so conspicuous, to give, in the proper place, a formal definition of the word. Its meaning appears more clearly from the fifth definition. This conjecture seems to be countenanced by the definitions of a straight line and a plane which stand in precisely the same predicament, no property of the line or plane being deduced from these definitions.
We may here remark generally, that although the definitions and propositions of the fifth book are expressed as if they applied only to magnitude, they are equally applicable to any other species of quantity. The student will find no difficulty in applying them to number, which is that species of quantity from which the clearest notions of proportion may be derived.
(447) IV. Magnitudes are said to have a ratio to one another,
when the less can be multiplied so as to exceed the other.
In order to have a ratio they must be • of the same kind,' and being so, one admits of being increased by multiplication so as to exceed the other.
[The student is advised to commence the propositions of the fifth book, omitting this and the succeeding definitions, and only to read them as he shall find them referred to from the propositions.]
(448) V. If there be four magnitudes, and any equimultiples
whatsoever of the first and third be taken, and also any equimultiples whatsoever of the second and fourth be assumed ; if, according as the multiple of the first is greater, equal to, or less than the multiple of the second, the multiple of the third is also greater, equal to, or less than the multiple of the fourth; then the first is said to have to the second the same ratio as the third has to
the fourth. The two magnitudes between which a ratio is conceived to subsist, are called the terms' of the ratio. That which is taken first in expressing the ratio is called the ' antecedent,' and the other is called the
consequent.' To express the ratio both these terms are used, and the sign : is commonly placed between them. Thus if A be the antecedent, and B the consequent, the ratio is expressed by A : B. The
А ratio of A to B is also more compendiously expressed thus
B It is evident from all that has been observed, that a ratio depends on the relative and not on the absolute magnitudes of its terms, and that therefore, although the terms be changed, it is possible that the ratio may remain the same. In other words, the same ratio may subsist between different pairs of magnitudes. The object of the preceding definition is to establish a criterion by which two ratios may be determined to be equal, and the selection of a proper criterion for this equality has given rise to much discussion among geometers. Without entering into the metaphysics of this subject, we shall