Pantometria — A Geometrical Treatize of Measuring (1591) - Page 16

Example.

A geometric diagram labeled with points A, B, C, D, E, and F, showing two triangles (ABC and DEF) used to illustrate the construction of a triangle with equal angles and side lengths.

The assigned triangle is A B C. To its angle at B, I frame an equal one contained by the lines D E F, extending E D and E F until they become of equal length with A B and B C. I can easily do this by extending my compass first to the lengths of A B and B C, and after transferring those distances to D E and E F. Finally, applying my ruler to the ends or limits of those lines, I draw the subtending side D F; thus, I have framed a new triangle D E F equal to the other, A B C.

I think it not amiss, before I treat of geometrical measurements, to present certain theorems whereby the ingenious may readily conceive the ground, reason, and demonstration of such rules as shall follow.

The First Theorem.

Any two straight lines crossing one another make the opposite or vertical angles equal.

The Second Theorem.

If any straight line falls upon two parallel straight lines, it makes the outward angle on the one equal to the inward angle on the other, and the two inward opposite angles on contrary sides of the falling line also equal.

The Third Theorem.

If any side of a triangle is extended, the exterior angle is equal to the two interior opposite angles; and all three angles of any triangle joined together are equal to two right angles.

The Fourth Theorem.

In equiangular triangles, all their sides are proportional, as well those that contain the equal angles as their subtending sides.

The Fifth Theorem.

If any four quantities are proportional, the first multiplied by the fourth produces a quantity equal to that which is made by the multiplication of the second by the third.

The Sixth Theorem.

The visible beams falling on plane, convex, or concave glasses are reflected in equal angles.

The Seventh Theorem.

In right-angled triangles, the square of the side subtending the right angle is equal to both the squares of its containing sides.

The Eighth Theorem.

All parallelograms are double the triangles that are described upon their bases, provided their altitudes are equal.

The Ninth Theorem.

All like or equiangular figures retain double the proportion of their corresponding sides.

The Tenth Theorem.

If from any angle of a triangle to its subtending side a perpendicular descends, the square of that subtending side (or base) added to one of the containing sides...