Dividing a Semicircle into Curvilinear 'Pyramids'
Taking a quarter of the half-circle by fractions of sixteen segments
The transcription continues a quadrature exercise in which the circle is divided into 16 curvilinear 'pyramids', of which every 3 compose one whole part, so that 4 whole parts (12 pyramids, the three-quarters) are drawn off from the half-circle, leaving 4. A worked fraction routine finds of what number 4 is the three-quarters, giving 16/3 in three different methods. Lettered sketches at the lower right (a b c; g d c f) show how a single curvilinear pyramid is halved along its base by a line of middle curvature.
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The sixteen pyramids and the three-quarters to be removed
The circle is divided into 16 curvilinear pyramids, and every 3 of them compose one whole part. Four whole parts carry 12 pyramids, the three-quarters of the whole, which are drawn off, leaving 4 pyramids, a quarter of the sixteen.
Three ways to find the number of pyramids
To learn of what number 4 is the three-quarters, multiply 4 by 4 to get 16, divide by the 3 above the line to get 5 and 1/3, then reduce to thirds to reach 16/3. A shorter method simply adds one third of 4 (one and 1/3) to reach 5 and 1/3, then makes all thirds.
Halving a curvilinear pyramid along its base
To divide a curvilinear pyramid into two equal parts along its length, dividing the base in half, set the circle of the lesser curvature within that of the greater curvature c and take the line b as a line of middle curvature. Placed at g c, between g d and g f, it cuts the pyramid into two equal parts upon the base d f, at the point c.
Equal side-curvatures require an equal middle curve
When the curvatures of the sides of the curvilinear pyramid are equal, the curvature of the middle line must be of the same curvature as the others.
