Notes from Mathilde Scholl 1904–1906

GA 91 — 4 November 1904, Berlin

3. Trigonometry

Triangles that can be superimposed exactly on top of each other are called congruent. If they cannot be superimposed exactly on top of each other, but can be superimposed in such a way that one can be placed on top of the other, but not of the same size, they are called similar if the third lines are parallel and partially overlap.

One side five units long. One side seven units long.

To get a similar triangle, I connect the third point and the fifth point, then I get two parallel lines.

Two triangles are similar if they can be superimposed on each other, coinciding at an angle and with two sides parallel.

Two triangles are similar if they have the same angles. In similar triangles, the sides opposite the same angle are in the same proportions.

(2:5 = 4:10) is a proportion
(a:b = c:d)

If you multiply the two outer or the two inner terms, you get the same product: Then (a \times d = b \times c). Lemniscate (multiplication) Circle (Division)

(a + b = 90°)
(c + d = 90°)
(d + b = 90°)
(c + a = 90°)

The square of the perpendicular is equal to the product of the two parts of the hypotenuse.

(ABD ~ BDC)
(AD:BD = BD:DC)
(BD \times BD =AD \times DC)
(P^2 = d \times D)
(P^2 = d' \times D')

You can cross the diameter on the circumference three times and then there is a remainder, this remainder is approximately (\frac{1}{7}) of the diameter. So the circumference of the circle is (3 \frac{1}{7}) times the diameter.

(U = 3 \frac{1}{7})
(U = 3,14159[D] = \pi[D])