Epitome of Ptolemy's Almagest (1496) - Page 17

First Book

I have chosen to present Ptolemy’s preface word-for-word, both because of the many most worthy opinions it contains, and because of Ptolemy’s authority, so that our imitation of him might be rendered more faithful. Now, let us happily descend to the science of chords.

First Proposition.

Because the proportion that the whole $z.g$ has to the line segment $g.d$ is the same as the proportion produced by $z.d$, the lines will be proportional, as is clear through the second part of the 16th proposition of the sixth book The author refers to Euclid’s Elements, Book VI, Proposition 16, which deals with the sides of proportional rectangles..

Given the diameter of a circle: to find the sides of the decagon, hexagon, pentagon, square, and equilateral triangle inscribed in the same circle.

Equilateral. a thing of equal sides The "penultimate of the first" refers to the angle which a side opposes: the square produced by one side of a triangle multiplied by itself is equal to the two squares described by the two remaining sides. This is clear through the penultimate proposition of the first book This is the Pythagorean Theorem, Euclid’s Book I, Prop. 47.. A geometric diagram of a semicircle. The diameter is a horizontal line labeled a-d-g, where d is the center. A vertical radius d-b is drawn. Point e is the midpoint of radius d-g. A line segment e-b is drawn. A point z is marked on the extension of segment a-d-g beyond d.

Given the diameter of a circle: to find the sides of the decagon, hexagon, pentagon, square, and equilateral triangle inscribed in the same circle.

¶ Let there be a semicircle a.b.g. erected above the diameter a. d. g. and center d. I shall draw the line d. b. perpendicular to a. g. through the 11th proposition of the first book of Euclid. I will then divide the line d. g. into two equal parts at point e. and draw the line e. b. I shall make e. z. equal to this line, and through the resulting line b. z., I say that z. d. is equal to the side of a decagon, and b. z. is equal to the side of a pentagon. I will show it thus: Because g. d. is divided into two equal parts at e. and the length d. z. is added to it, therefore, through the sixth proposition of the second book, the rectangle formed by g. z. and d. z. together with the square of d. e. is equal to the square of the line e. z. But e. z. is equal to e. b., and through the penultimate of the first book, the square of e. b. is equal to the two squares of b. d. and d. e. Therefore, the rectangle formed from g. z. and d. z. with the square of d. e. will be equal to the two squares of b. d. and d. e. By subtracting the common square of d. e., the rectangle formed by g. z. and d. z. will be equal to the square of b. d., and thus also equal to the square of d. g. Therefore, by the second part of the 16th proposition of the sixth book, the proportion of g. z. to d. g. will be as the proportion of d. g. to d. z. Thus, by the principle of the sixth book, the line z. g. is divided at point d. according to the ratio having a mean and two extremes This refers to the Golden Ratio.. But its larger portion, namely d. g., is the side of a hexagon through the corollary of the 16th proposition of the fourth book. Therefore, by the converse of the ninth of the thirteenth book, its smaller portion, namely d. z., is the side of a decagon—which is the first part.

number g. 13th book A line is said to be divided according to the ratio having a mean and two extremes when the proportion of the whole to its larger section is the same as that of the larger to the smaller, as is clear through the principle of the 6th book.

¶ And since, by the penultimate of the first book, the square of b. z. is equal to the two squares of b. d. and d. z., and b. d. is the side of the hexagon and d. z. is the side of the decagon, therefore, by the converse of the tenth of the thirteenth book, b. z. will be the side of the pentagon—which is the second part. ¶ If you draw line a. b., it will be established from the sixth of the fourth book that it is the side of a square inscribable in a circle. Moreover, through the eighth of the thirteenth book, it is manifest that the side of a triangle is "triple in power" original: "potentialiter triplum," meaning the square of the side of the triangle is three times the square of the radius. to the side of the hexagon or the semi-diameter. Therefore, by whatever division the diameter has been divided, in that same division its half will be established (namely the side of the hexagon); the square of this half and the square of the other half are the square of the line z. e. Therefore, z. e. is known, and once d. e. is subtracted from it, z. d. will remain known: the chord of the tenth part of the circle A 36-degree arc.. But also, the square of this together with the square of the side of the hexagon are the square of the side of the pentagon; thus the chord of the fifth part of the circle A 72-degree arc. becomes known. The square of the side of the square original: "tetragoni" is double the square of the side of the hexagon; and the square of the side of the triangle original: "trigoni" is triple the same square of the side of the hexagon; therefore both of them will become known.

If the side of the hexagon is equal to the side, then the side of the decagon and the side of the hexagon joined together in a straight line will produce a total line divided by the mean and extreme ratio; its larger portion is the side [of the hexagon], as is clear through the ninth of the 13th book.

Second Proposition.

Given the chord of any arc: the chord of the remaining arc of the semicircle becomes known.

¶ It is clear from the 30th proposition of the third book that the angle contained by such chords is a right angle. Therefore, through the penultimate of the first book, the square of the diameter of the circle will be equal to the two squares of those chords; therefore, etc. Thus, from the side of the decagon, you will find the chord of an arc of 144 degrees.

Note [mathematical scribbles] The square which is described from the side of a right-angled triangle opposite the [right] angle, multiplied by itself, is equal to the two squares which are constituted from its two [other] sides. A circular arc diagram illustrating various chords extending from a common vertex at the bottom-right. The arcs subtended by these chords are labeled with degree measurements: 60, 72, 90, 108, 120, and 144. There are numerous handwritten labels around the arc and its chords. Diameter: 120 ¶ The square of the diameter is equal to the squares of the two chords of the right angle.