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HEnce you have a further Confirmation of Consect. 2. Prop. 32. and Prop. 36. N. 3.
II. Hence also naturally flows a Confirmation of Consect. 2. Definit. 20. and consequently the Dimension of the Sphere both as to its solidity and Surface. For putting a for the Dia∣meter of the Sphere and circumscribed Cylinder, and Ea for the Circumference, the Basis of the greatest Circle will be ¼ eaa, and ••hat multiplied by the Altitude, gives ¼ ea{powerof3} for the Cylinder. Therefore by the present Proposition, ⅙ ea{powerof3} gives the Solidity of the Sphere (by making as 3 to 2 so ¼ to ⅙) This divided by ⅙ a, will give, by vertue of Consect. 1. of the aforesaid Def. 20. and Consect. 3 Definit. 17. the Surface of the Sphere eaa.
III. Therefore the(α) 1.1 Surface of the Sphere eaa, is manifest∣ly Quadruple of the greatest Circle ¼ eaa.
IV. The Surface of the Cylinder, without the Bases, made by multiplying the Altitude a by the Circular Periphery of the Base ea, will be eea, equal to the Surface of the Sphere.
V. Adding therefore the 2 Bases, each whereof is ¼ eaa, the whole Surface of the Cylinder 1 ½ eaa, will be to the Sur∣face of the Sphere eaa as 3 to 2.
VI. The Square of the Diameter aa to the Area of the Cir∣cle ¼ eaa, is as a to ¼ ea, i. e. as the Diameter to the 4th part of the circumference.
VII. A Cone of the same Base and Altitude with the Sphere and Cylinder, will be by Consect. 2. of this, Prop. and the Con∣sect. of Prop. 38. 〈 math 〉〈 math 〉 ea{powerof3}, and of the Cylinders ¼ or 〈 math 〉〈 math 〉 ea{powerof3}. There∣fore a Cone, Sphere, and Cylinder, of the same heighth and dia∣meter, are as 1, 2, 3. The Cone therefore is equal to the Excess of the Cylinder above the Sphere; as is otherwise evident in Scholium 1. of this.