Demonstration.
For, 1. the Bases of 2 similar Triangles or Parallelograms
Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.
About this Item
- 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.
- Publication
- 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.
- Link to this Item
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http://name.umdl.umich.edu/A61912.0001.001
- Cite this Item
-
"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 111
for which any 2 Homologous Sides, e. g. AB and AB (Fig. 75.) may be taken) and Perpendiculars let fall thereon DE and DE, will be by Consect. 1. Prop. 34, as a to ea, b to eb. Therefore the Parallelograms and Triangles themselves, will be as ba to eeba, by Consect. 7 and 8. Def. 12. i. e. by Def. 34. in duplicate Reason of their Perpendiculars or assumed Sides, which is most conspicuous in Squares, which putting a for the ••ide of one, and ea for the other, are to one another as aa to ••eaa.
2. Like Polygons are resolved into like Tri∣••ngles, when the Triangles ABC and ABC,(α) 1.1 and ••lso AED and AED are Equiangular, by Con∣sect. 3. Prop. 34. but CAD and CAD, are also Equiangular, because each of their angles are the remainder of ••qual ones, after equal ones are taken from them. Wherefore ••he first Triangles are in duplicate Proportion of the sides BC ••nd BC; the second likewise of the sides CD and CD the ••hird are also in the same Proportion of the sides DE and DE, &c. i. e. (since by the Hypoth. BC has the same reason to BC ••s CD to CD, and DE to DE) each to each is in duplicate Pro∣portion of the sides BC to BC, or CD to CD, by the first of ••his. Therefore by a Syllepsis, the whole Polygons are in du∣plicate Proportion of the same Sides: Which is the second thing ••o be demonstrated.
3. Circles and their like Sectors, are as the Squares of their Diameters, by Prop. 32. therefore in duplicate Proportion of them, by the first of this: Which is the third thing: Therefore simi∣••ar Plane Figures, &c. Q. E. D.
Notes
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(α) 1.1
Eucl. 19 & 20 lib. 6.