Pyramids cut by parallel planes; triangle centres of gravity

Sheet 526a Recto

Right-hand column.

Pyramids cut by lines equidistant from their base will always keep within themselves the same proportion among their sides that the whole pyramid had with its entire sides.
* The upper part of the pyramid cut by a line equidistant from its base ...
* Such shall be the proportion of the sides of the upper part of the cut pyramid as is the proportion that ...
* Such shall be the proportion the parts of the sides have among themselves with the part of the cut pyramid, as was that of the whole sides with the whole of the pyramid.
* The part of the pyramid cut equally distant from its base will have such proportion within itself among its sides as was that which the sides of the whole pyramid had among themselves.
The pyramid ...
As was that which the sides of the whole pyramid had among themselves ...
The part of the pyramid cut equally distant from its base will have such proportion within itself among its sides as was that which the sides had among themselves. The sides of the part of the cut pyramid have among themselves the same proportion the sides of the whole pyramid had among themselves.
The sides of the pyramid cut equally distant from its base never vary their proportion.

Triangle with the letters:

c - f e - b a

It is proved. Let the cut of the pyramid be at e f. I say that, because the whole pyramid has its base half of its axis, the base ef does the same in its cut portion. And if you should wish to say no, then prove it.

Central column.

* Every triangle has within it 3 medians, and each median divides it into two equal parts, whence in every median is found the centre of gravity of the whole triangle; whence 3 centres are shown, and since this centre cannot be other than one and in a single place, it must lie at the intersection of the said medians.
* 3 are the medians of every triangle, and each median divides it into 2 equal parts, whence each of them has within it the centre of gravity of the whole triangle. Therefore we find in triangles 3 centres of all its gravity; whence it is concluded that, since it cannot be other than one and in a single place, it must of necessity lie at the intersection of the said medians.

Within the preceding text, a figure of a triangle with:

* A median is that which passes ...
* The centre of gravity of every triangle will be at the intersection of its medians.
This is proved by alleging that every median has within it the centre of gravity of that triangle. And the medians are 3, and the centre of the triangle is but one; therefore it must lie at the intersection of these medians, because in that place each one touches it; whence we shall say there are 3 centres reduced to one.
It is proved. Every median divides the triangle into two parts of equal weight. Therefore it has within it the centre of gravity of the whole triangle; and the medians are 3 and each has within it the said centre, so here there would be 3 centres of gravity; but since nature gives only one, each median must carry its centre of gravity in one and the same place. Therefore necessity places it at their intersections.
* It is proved. Every median divides the triangle into 2 parts of equal weight. Therefore that median has within it the centre of gravity of the whole triangle. And the medians are 3, and it follows there are 3 centres, and the triangle has but one. Whence it is concluded to lie, of necessity, at the intersection of the said medians.

Left-hand column.

The centre of every triangle that has its upper angle by a perpendicular line above one angle of its base will have the centre of gravity in the middle of its length.

A suspended triangle, with:

b - m o n - a

* a b cuts the pyramid into 2 equal parts, and n m also cuts that pyramid. Therefore at the point o of the line n m the middle of the weight of the triangle of the line a b is met.
* Every triangle is divided by 3 medians, and each median has within it the centre of gravity of the whole triangle.

A pulley applied to a sphere, with:

b a

* A way of weighing any portion of a sphere without removing it from that sphere, and of knowing what proportion it has to the whole sphere.
* And 3 medians hold within them 3 centres, and the triangle has but one. Therefore it must lie at the intersection of these medians.
* Every median divides the triangle into 2 parts of equal weight. Therefore that median contains within it the centre of gravity of the whole triangle, and so ...
* Therefore, the triangle having within it 3 medians with 3 centres, and the ...
* Every triangle has within it 3 medians, and each by itself divides that triangle into 2 equal parts, whence each has within it the centre of gravity of the whole triangle; this being so, it is ...

In the margin, a sectioned cube or parallelepiped, with:

2-2

Sheet 526b Recto

At the top, to the right.

From an endpoint of a given line, to make a right angle.

A right triangle within a semicircle, with:

a - c d b

Let a c be the given line, from which the equilateral triangle a b c will be made; and with the same opening of the compass the point will be set at b and the semicircle c a d will be drawn, and the line d a will be drawn, and a will be a right angle.

On the left, a rectangle on a circular sector, with:

b - n - d c a

The whole a b n is equal to the whole n c d. Take away n from each, and a b remains equal to c d. Take b from a, and a remains alone. Take c from d, and d remains alone, equal to a. On one side there remains a n b, from which if a is taken away, n b remains; on the other side if a is taken away, c n remains equal to b n.

Lower half, from right to left. A figure with:

n - o

o n are the wholes.

A half-column within a rectangle. A double figure with:

b; a
a b equal.

A figure with:

b - a

a b equal.

A figure with:

c — d

c d equal.

A figure with:

f — e

e f equal.

A figure with:

p — o

o is squared, p s ...

Central half-column. A double figure with:

a; b
a b equal.

A figure with:

f e - b

b e f equal.

A broken figure and a figure with:

f

There remains to square the part f, as on the other side.

On the left, a figure with:

c a – d b

On the right, toward the margin:

Tell [me].

Foglio 526 a Recto

Colonna destra.

Triangolo colle lettere:

Colonna centrale.

Entro lo scritto precedente, figura di triangolo con:

Colonna sinistra.

Triangolo sospeso, con:

Carrucola applicata a una sfera, con:

Nel margine, cubo o parallelepipedo sezionato, con:

Foglio 526 b Recto

In alto, a destra.

Triangolo rettangolo entro un semicerchio, con:

A sinistra, rettangolo su settore di cerchio, con:

Metà inferiore da destra a sinistra. Figura con:

Semicolonna entro un rettangolo. Doppia figura con:

Figura con:

Figura con:

Figura con:

Figura con :

Semicolonna centrale. Doppia figura con:

Figura con:

Figura spezzata e figura con:

A sinistra, figura con:

A destra verso il margine:

Sheet 526a Recto

Right-hand column.

Triangle with the letters:

Central column.

Within the preceding text, a figure of a triangle with:

Left-hand column.

A suspended triangle, with:

A pulley applied to a sphere, with:

In the margin, a sectioned cube or parallelepiped, with:

Sheet 526b Recto

At the top, to the right.

A right triangle within a semicircle, with:

On the left, a rectangle on a circular sector, with:

Lower half, from right to left. A figure with:

A half-column within a rectangle. A double figure with:

A figure with:

A figure with:

A figure with:

A figure with:

Central half-column. A double figure with:

A figure with:

A broken figure and a figure with:

On the left, a figure with:

On the right, toward the margin:

On this page

A truncated pyramid keeps the proportion of the whole

A triangle's centre of gravity at the crossing of its medians

A pulley for weighing a portion of a sphere

A right angle raised from the end of a line

Quadrature exercises equating lettered figures