28. If unequal parts be taken from two equal quantities, and betwixt the whole and the part of each there be interposed two means, one in geometrical, the other in arithmetical proportion; the difference betwixt the two means will be greatest, where the difference betwixt the whole and its part is greatest. |A E G H B
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A F I K B|For let A B and A B be two equal quantities, from which let two unequal parts be taken, namely, A E the less, and A F the greater; and betwixt A B and A E let A G be a mean in geometrical proportion, and A H a mean in arithmetical proportion. Also betwixt A B and A F let A I be a mean in geometrical proportion, and A K a mean in arithmetical proportion; I say H G is greater than K I.
| For in the first place we have this analogism | A B. A G :: B G. G E, by [article 18]. |
| Then by composition we have this | A B + A G. A B :: B G + G E that is, B E. B G. |
| And by taking the halves of the antecedents this third | ½A B + ½A G. A B :: ½B G + ½G E, that is, B H. B G. |
| And by conversion a fourth | A B. ½A B + ½A G :: B G. B H. |
| And by division this fifth | ½A B - ½A G. ½A B + ½A G :: H G. B H. |
| And by doubling the first antecedent and the first consequent | A B - A G. A B + A G :: H G. B H. |
| Also by the same method may be found out this analogism | A B - A I. A B + AI :: K I. B K. |
Now seeing the proportion of A B to A E is greater than that of A B to A F, the proportion of A B to A G, which is half the greater proportion, is greater than the proportion of A B to A I the half of the less proportion; and therefore A I is greater than A G. Wherefore the proportion of A B - A G to A B + A G, by the [precedent lemma], will be greater than the proportion of A B - A I to A B + A I; and therefore also the proportion of H G to B H will be greater than that of K I to B K, and much greater than the proportion of K I to B H, which is greater than B K; for B H is the half of B E, as B K is the half of B F, which, by supposition, is less than B E. Wherefore H G is greater than K I; which was to be proved.
Coroll. It is manifest from hence, that if any quantity be supposed to be divided into equal parts infinite in number, the difference between the arithmetical and geometrical means will be infinitely little, that is, none at all. And upon this foundation, chiefly, the art of making those numbers, which are called Logarithms, seems to have been built.
29. If any number of quantities be propounded, whether they be unequal, or equal to one another; and there be another quantity, which multiplied by the number of the propounded quantities, is equal to them all; that other quantity is a mean in arithmetical proportion to all those propounded quantities.
CHAP. XIV.
OF STRAIT AND CROOKED, ANGLE AND
FIGURE.
[1.] The definition and properties of a strait line.—[2.] The definition and properties of a plane superficies.—[3.] Several sorts of crooked lines.—[4.] The definition and properties of a circular line.—[5.] The properties of a strait line taken in a plane.—[6.]. The definition of tangent lines.—[7.] The definition of an angle, and the kinds thereof.—[8.] In concentric circles, arches of the same angle are to one another, as the whole circumferences are.—[9.] The quantity of an angle, in what it consists. —[10.] The distinction of angles, simply so called.—[11.] Of strait lines from the centre of a circle to a tangent of the same. —[12.] The general definition of parallels, and the properties of strait parallels.—[13.] The circumferences of circles are to one another, as their diameters are.—[14.] In triangles, strait lines parallel to the bases are to one another, as the parts of the sides which they cut off from the vertex.—[15.] By what fraction of a strait line the circumference of a circle is made.—[16.] That an angle of contingence is quantity, but of a different kind from that of an angle simply so called; and that it can neither add nor take away any thing from the same.—[17.] That the inclination of planes is angle simply so called.—[18.] A solid angle what it is.—[19.] What is the nature of asymptotes.—[20.] Situation, by what it is determined.—[21.] What is like situation; what is figure; and what are like figures.
The definition end properties of a strait line.
1. Between two points given, the shortest line is that, whose extreme points cannot be drawn farther asunder without altering the quantity, that is, without altering the proportion of that line to any other line given. For the magnitude of a line is computed by the greatest distance which may be between its extreme points; so that any one line, whether it be extended or bowed, has always one and the same length, because it can have but one greatest distance between its extreme points.