By Jacques Hadamard Translated by Mark Saul
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Nonetheless, a hundredth of a grad is often called a centesimal minute, and is indicated by the symbol ` (to distinguish it from the hexagesimal minute, which is a sixtieth of a degree). Likewise, one hundredth of a centesimal minute is called a centesimal second , denoted by the symbol ``. 54`17``. 9 400 G ◦ = A grad is equal to 360 400 , that is 10 of a degree, or 54 . A degree equals 360 of a grad). 11`11``1 . . (in other words, 10 9 19. Theorem. Given a line, and a point not on the line, there is one, and only one, perpendicular to the line passing through the given point.
Then, in all the equations we write, the quantities which appear on the two sides of an equality will represent not the quantities themselves, but rather their measures. This will allow us to write a number of equations which otherwise would have no meaning. For instance, we might equate quantities of different kinds, since we will be dealing only with the numbers which measure them, the meaning of which is perfectly clear. We will also be able to consider the product of any two quantities, since we can talk about the product of two numbers, etc.
The theorem is proved. If n is the number of sides of the polygon, the sum of the angles is 2(n − 2) or 2n − 4 right angles. Corollary. The sum of the exterior angles of a convex polygon, formed by extending the sides in the same sense, is equal to four right angles. Indeed, an interior angle plus the adjacent exterior angle gives us two right angles. Adding the results for all the n vertices we obtain 2n right angles, of which 2n − 4 are given by the sum of the interior angle. The sum of the exterior angles is equal to the four missing right angles.