Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

Demonstration.

For having divided the circumference of the circle first into three equal parts, there will be drawn to the second spiral four right lines BE, BG, BI and BA being as 3, 4, 5, 6, and but only three sectors circumscrib'd, viz GBg, IBi and ABa, which proceed according to the squares of the three latter lines, viz. 16, 25, 36, so that the sum is 77, while the sum of three equal to the greatest is 108, and so the one to the other (dividing both sides by 9) as 12 to 8 〈 math 〉〈 math 〉▪ Having moreover bisected the arches and parts of the line BE, so that that shall be 6, the second BF will be 7, and so the other five 8, 9, 10, 11, 12; and the sectors answering to them (excepting the first) 49, 64, 81, 100, 121, 144, so that their sum shall be 559, while the sum of six equal to the greatest, i. e. the whole circle is 864, and so one to the other (dividing both by 72) as 12 to 7 〈 math 〉〈 math 〉. In the other bisection of the arches and the parts of the line BE, so that the one shall be 12, the second 13, &c. to the thirteenth BA which will be 24, the sum of twelve sectors will be found to be 4250; and the sum of

as many equal to the greatest 6912, and so the one to the other (dividing both sides by 576) as 12 to 7 〈 math 〉〈 math 〉 s. 〈 math 〉〈 math 〉. Therefore the proportion will be

• I. In the first case 12 to 7+1+½+〈 math 〉〈 math 〉 viz. 〈 math 〉〈 math 〉.
• II. In the second case 12 to 7+½+¼+〈 math 〉〈 math 〉 viz. 〈 math 〉〈 math 〉.
• III. In the third case 12 to 7+¼+⅛+〈 math 〉〈 math 〉, &c.

The first and second fractions thus decreasing by ½ the latter by ¼. Wherefore the proportion of the second circle to the second spiral space will be as

12 to 7+1+½+〈 math 〉〈 math 〉

−½−¼−〈 math 〉〈 math 〉

−¼ &c.−⅛ &c.−〈 math 〉〈 math 〉 &c.

By vertue of Consect. 3. and 8.=0=0 Prop. 21. lib. 1. i. e. as 12 to 7. Q. E. D.