Proposition XLIX.
THe side (AB) of an Equilateral Triangle (ABC, Fig. 92. N. 1.) inscribed in a(α) 1.1 Circle, is in Power triple of the Radius (AD) i. e. of the □ of AD.
Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.
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- Title
- Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.
- Author
- Sturm, Johann Christophorus, 1635-1703.
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- London :: Printed for Robert Knaplock and Dan. Midwinter and Tho. Leigh,
- 1700.
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- Subject terms
- Mathematics -- Early works to 1800.
- Geometry -- Early works to 1800.
- Algebra -- Early works to 1800.
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http://name.umdl.umich.edu/A61912.0001.001
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"Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S." In the digital collection Early English Books Online 2. https://name.umdl.umich.edu/A61912.0001.001. University of Michigan Library Digital Collections. Accessed June 20, 2025.
Pages
Page 138
Demonstration.
MAke AD or FD=a, and so its Square aa. Since There∣fore, having drawn DF thro' the middle of AB, or the middle of the Arch AFB let DE be = ½a; for the angles at E are right ones, by Consect. 5. Definit. 8. and the Hypothe∣nuses AD, AF, are equal, by Schol. of Definition 15. but the side AE is common. Therrefore the other Sides FE and ED are equal by, by Consect. 3. Prop. 43.) and the □ of the latter is ¼ aa, which subtracted from aa leaves ¾ aa for the □ of AE. Therefore the line AE is 〈 math 〉〈 math 〉, and conse∣quently AB 〈 math 〉〈 math 〉, i. e. 〈 math 〉〈 math 〉, i. e. 〈 math 〉〈 math 〉: therefore □ AB=3aa. Q E. D.
CONSECTARYS.
I. IF the Radius of a Circle be = a, the side of an Inscribed Regular Triangle will be 〈 math 〉〈 math 〉, e. g. if AD be 10, AB will be 〈 math 〉〈 math 〉; and if AD be 10,000,000, AB will be 〈 math 〉〈 math 〉, i. e. 17320508, and the Perpen∣dicular DE 5000,000.
II. Hence it is evident, that in the genesis of a Tetraedrum pro∣posed in Def. 22, that the elevation CE (Fig. 44, N. 1.) is to the remaining part of the Diameter of the Sphere CF as 2 to 1; for making the Radius CB=a and its □ aa, the □ of AB or BE will =3aa, by the present Proposition. Therefore the □ of CB being subtracted from the □ of BD or BE, there remains the □ of CE=2aa. But since CE, CB, CF, are continual Proportionals, by N. 3. Schol. 2. Prop. 34. CE will be to CF as the □ of CE to the Square of CB, by vertue of Prop. 35. i. e. as 2 to 1.
SCHOLIUM.
HEnce you have the Euclidean way of generating(α) 1.2 a Te∣traedrum, and inscribing it in a given Sphere, when he bids you divide the Diameter EF of a given Sphere so that EC shall be 2 and CF 1, and then at EF to erect the Perpendicular CA, and by means thereof to describe the Circle ABD, and to inscribe therein an Equilateral Triangle, &c.
Notes
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(α) 1.1
Ptol. lib. I. Almagesh.
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(α) 1.2
Eucl. 12 lib. 12.