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Title:  A treatise of angular sections by John Wallis ...
Author: Wallis, John, 1616-1703.
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LIII. But, The subtenses of the Quintant and Sesquiquintant, (that is, of 72, andFig. XXVIII. of 108 Degrees, which together complete the Semicircumference) Multiplied the one into the other, (or the Rectangle of them,) divided by the Radius; is equal to the Subtense of the double Arch of either. For, by § 9, Chap. R) AE (B. That is, of 144, or of 216 Degrees. That is, of the double, or Triple Quintant, (these two having the same Subtense.) That is, . That is,LIV. The Radius Multiplied into, is equal to the Subtense of the Biquin∣tant, and of the Triquintant; That is, to the Subtense of 144, and of 216 De∣grees.LV. And the Square of this subtracted from the Square of the Diameter, leaves the Square of the Subtense of 36 Degrees; (as being what 144 Degrees wants of a Semicircumference, and what 216 exceeds it. For 180 − 144 = 36 = 216 − 180.) And the Square Root thereof is that Subtense, . That is,LVI. The greater Segment of the Radius cut in extream and mean Proportion, is the Subtense of 36 Degrees. That is, of half a Quintant, or the side of the inscri∣bed Decagon.LVII. But we had before shewn (at § 38.) that this added to the Radius (which is the Subtense of 60 Degrees, or side of the inscribed Hexagon,) is equal to the Subtense of 108 Degrees, or Sesquiquintant: Therefore,LVIII. The Aggregate of the subtenses of 36 Degrees, and of 60 Degrees, (that is, the sides of the inscribed Decagon and Hexagon,) is equal to that of 108 Degrees; (that is, of the Sesquiquintant, or three Tenths.)LIX. If therefore to the Subtense of 36 Degrees, , be added that of 108 Degrees , it makes , or . That is,LX. The Subtense of the Semiquintant (or 36 Degrees) and of the Sesquiquintant (or 108 Degrees) added together, are in power Quintuple to the Radius, (that is, the Square of that Aggregate is equal to five Squares of the Radius.) For, .LXI. And their Difference is equal to the Radius. For, .LXII. And the Rectangle of them, is equal to the Square of the Radius. For, .LXIII. And the sum of their Squares is Triple to the Square of the Radius. (Or, equal to the Square of the side of the inscribed Trigone.) That is, .0