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

Proposition XXXVIII.

A Parallelepiped(α) 1.1 BF (Fig. 77. N^o 3.) is to a Pyra•••••• ABCDE upon the same Base BD and of the same heighth, 〈◊〉〈◊〉 3 to 1. This was D••monstrated in Consect. 3. Definit. 17. bu•• here we shall give you another.

Demonstration.

Suppose 1 the whole Al••itude BE divided into 3 equal Pa•••• by transverse P••ains Parallel to the Base, then will (by reason 〈◊〉〈◊〉 the Similitude of the Pyramids abcd ACBD•• and AECDE) the Bases abcd ABCD and ABC•• be by Consect. 2. Prop. 34 and Consect. 3. Defi•••••• 17 in duplicate Proportion of the Altitudes i. e. in duplicate Arithmetical Progression 1, •• 9, mo••eover 2, bisecting the parts of the Altitude, the qu••¦d••angular Sections now double in Number (as the Indivisibles o•• Elements of the proposed Pyramid) will be as 1, 4, 9, 16, 2••

36, &c. ad Infinitum, all along in a duplicate Arithmetical Pro∣portion; while in the mean time there answer to them as many Elements in the Parallelepiped equal to the greatest ABCD, wherefore by Consect. 10. Prop. 21. all the Indivisibles of the Pyramid taken together will be to all the Indivisibles of the Parallelepiped also taken together, i. e. the Pyramid it self to the Parallelepiped, as 1 to 3. Q. E. D.

CONSECTARY.

THis Demonstration may be easily accommodated to all other Pyramids and Prisms, and also Cones and Cylinders,(α) 1.2 since here also (Fig. 78.) the circular Planes ba, BA, and BA are as the squares of the Diameters, and so as 1, 4, 9. and so likewise all the other Elements of the Cone by continual bisection are in duplicate Arithmetical Progression; when in the mean time there answer to them in the Cylinder as many Ele∣ments equal to the greatest BA &c.

Notes

• (α) 1.1

  Euclid. Prop 7 Coroll. lib. 12.
• (α) 1.2

  Euclid. Prop. 10. lib. 12.