Notes from Mathilde Scholl 1904–1906

GA 91 — 19 October 1904, Berlin

1. The Difference Between Calculation [and] Operation

If we conceive of the relationships between quantities, we have general arithmetic. Special arithmetic deals with specific quantities of units. Algebra is named after “Algeber” (an Arabic mathematician).

First relation between numbers: addition. One number is added to another: (a + b) and so on. This is extended to a series:

(a + b + c + d + e + f) and so on.

Second relation:

subtraction; you take one number away from another: (a - b). This expands: (a - b - c - d - e) and so on.

One prerequisite is necessary here, namely that (a) is greater than (b).

(a \gt b) (then (a) is greater than (b))
(a = b) (then (a) is equal to (b))
(a \lt b) (then (a) is less than (b))
(c =) the result: (a - b = c).

If (a) is greater than (b), then the question is how much remains if I take away (b) from (a)? (c) remains. (c) is the number by which (a) is greater than (b). If (a) is equal to (b), then (c = 0). If (a) is smaller than (b), then there is also a (c), for example: (3 - 5 =?) Here you ask how many units are missing from (a) if (a) is not as large as (b).

[This means: For] a number that is minus, it means: There was a subtraction; in this subtraction, the minuend was too small. — A negative number can only be understood as the result of an arithmetic operation.

The addition can be such that some numbers are equal to each other or all are equal to each other.

(a + b + a + c) and so on, or
(a + a + a + a + a + a) and so on

If all numbers are equal in the addition, we have multiplication. The number that tells you how many times the same number is in the sum is called the multiplier.

(a \times b) means: (a) is to be added to (b) the same number of times. or (a \cdot b): (a) is multiplied by (b). (a \cdot b \cdot c \cdot d) (means: (a) is multiplied by (b), (c), (d) and so on one after the other).

When subtracting, you may subtract different numbers:
(a - b - c - d - e - f) and so on, or the same ones, for example (a - b - b - b - b - b) and so on.

The question is: how many times can I subtract b from a?
(a : b) is division (or a hidden subtraction).

The divisor is actually a minuend. The dividend is actually a subtrahend.

It is a special case of subtraction.

(a \cdot b \cdot c \cdot d \cdot e \cdot f)
(a \cdot a \cdot) and so on, m times (meaning: the same number multiplied over and over again by itself), is denoted by (a^m) ((a m) times)
(a \cdot a \cdot a \cdot a \cdot a \cdot a) and so on, multiplied by (m) (meaning: the same number multiplied over and over again by itself) is designated by (a^m) ( (a m) times)

(a^m = P) ((a m) times taken gives (P) = exponentiation).
(a) to the mth [power] (=) P

(a =) the root, the base or the factor.
(m =) the exponent.
(P =) the power.

The fifth arithmetic operation is exponentiation.

If P and m are known and a is sought:
(\sqrt[m]{P} = x) Square root.

abbreviated division:
(P) = radicand
(m =) root exponent
(x(a) =) radix.

If (m) is unknown, write (a^x = P).
To what power of a do you have to raise a to get (P)?
(x = log_a P) (the exponent in the parenthesis is the logarithm)
(2 = log_3 9) ((2) is equal to the log of (9) with respect to the root (3))

Taking logarithms is the seventh arithmetic operation.

(a \cdot b)
(+a \cdot +b)
(+a \cdot +a \cdot +a \cdot +a + b - times)
(+a \cdot +b = +ab)
(-a \cdot +b = -ab)
(+a \cdot -b = -ab)
(-a \cdot -b = +ab)

Multiplying equivalent numbers by each other yields a positive result; non-equivalent numbers yield a negative result.

(+a \cdot -b = -c)
(+a \cdot +b = +c)
(-a \cdot +b = -c)
(-a \cdot -b = +c)

Similar numbers divided by each other give a positive result, unlike numbers divided by each other give a negative result.

(a^b = c) where (+a) and (+c)

or (-a) and (+c)

(\sqrt[b]{c} = a)

((+a)^+b = +c)

((-a)^+b = +c)

To denote a negative power, imaginary numbers must be found that are neither positive nor negative.

A triangle is a figure bounded by three straight lines. We distinguish triangles according to the size of their sides and their angles. First, we distinguish a triangle in which all three sides are equal, calling it an equilateral triangle, which also has all three angles equal. A triangle in which only two sides are equal and the third unequal is called an isosceles triangle. It has two equal angles, which are adjacent to the unequal side. Then one, which is unequal; all sides and angles are unequal. Size ratios of the angles of a triangle.

2. Secondary angle

Measuring the angle: The angles are measured according to the relative size of the arc. You measure an angle by drawing a circle around it.

First divide the circle into four parts, then again into two parts, then each of these into five and each of these into (9) parts = (4 \times 2 \times 5 \times 9 = 360°) (degrees). The angle is as large as the angle is of the (360°).

Two minor angles always add up to (180°).

Two vertex angles.

You name the angle by writing a letter in it.

(\angle a) is called angle (a)

(\angle a = \angle b) as vertex angles

(\angle a + \angle b = 180°) as an angle between

Four pairs of opposite angles

(x) the intersecting line
(y), (z) the lines intersected

When two straight lines are intersected by a third, four pairs of opposite angles arise, namely those that lie on the same side of the intersecting line and on the same side of the intersected line.

Alternate angles are those that lie on different sides of the intersecting line and on different sides of the line being cut. Four pairs of alternate angles.

Angles are those that lie on the same side of the intersecting line and on different sides of the line being cut. Four pairs of angles.

(a) and (b) are opposite angles.

If two parallels are intersected by a third, then the opposite angles are all equal to each other.

(b) and (a) are alternate angles.

Assume that (b) and (a) are alternate angles at [two intersecting parallels].

Alternate angles between parallel lines must always be equal.

(b) and (a) alternate angles (=) assumption.

Proof:

((\angle b = \angle c))
(\angle b = \angle c) as opposite angle
(\angle a = \angle c) as vertex angleergo: (\angle a = \angle b)

If two quantities of a third are equal, they are also equal to each other.

This is an axiom, on which the proof is based. An axiom is a proposition that is neither capable nor in need of proof.

(a) and (b) are angles.

Proof:

(\angle a) and (\angle b) are angles (assumption)
(\angle a = \angle c) as opposite angle
(\angle b + \angle c = 180)° as adjacent angle
ergo: (\angle b + \angle c = 180°)
Proof that two angles are always (180°).

Axiom: In every calculation, one quantity can be replaced by another equal to it without changing the calculation.

(ABC) a triangle (assumption)
(DE || AC)
(\angle a = \angle d) as alternate angles
(\angle c = \angle e) as alternate angles
(\angle b = \angle b)

(\angle d + \angle b + \angle e = 180°)
ergo: (\angle a + \angle b + \angle c = 180°)

The sum of the three angles of any triangle is [(180°)].

(A B C D) quadrilateral. Given
(\angle e + \angle a + \angle g = 180°)
(\angle f + \angle h + \angle c = 180°)
(\angle c + \angle a + \angle g + \angle f + \angle h + \angle c = 360°)
(\angle a + \angle b + \angle d + \angle c = 360°)
ergo: In every quadrilateral, the sum of the (4) angles is (360°)

Line: spatial size of one dimension - length
Area: spatial size of two dimensions - length + width
Body: spatial size of three dimensions - length + width + thickness
? Spatial size of four dimensions

A two-dimensional space is bounded by a surface. A right angle has a quarter of a circular arc between its legs.

A square is a surface that is bounded by four equal lines and has four right angles.

Line with four units of measurement, (e) one unit of length, (ab = 4 \times e) ((4 \times) the unit), (A B = 4 \times e)

You can measure a length by counting how many times the unit is included.

(e) (size (e)) squared is the unit area. The unit area is a c described in terms of the unit length.

Area unit. Question: How much larger than the area unit is the whole square when the side is four times larger than the length unit?

(A B C D = AB^2)

The area of a square is found by multiplying one side by itself and raising it to the power of two.

(1^2 = 1)
(2^2 = 4)
(3^2 = 9)
(4^2 = 16)
(5^2 = 25)
(6^2 = 36)
(7^2 = 49)
(8^2 = 67)
(9^2 = 81)
(10^2 = 100)

The two sides of a right angle form the cathets. The opposite line is the hypotenuse.

(3^2 = 4^2 = 5^2)

(BDEC = BC^2)
(HEFJ = \angle ABC + HJCE)
(CE =BC)
(HEFJ = \angle ABC + HJCE)
(CE =BC)
(ABC = BDG)
(JKC = BGK)
(BDEC = BGHECB + BGD + HCE)
(HEFJ = HJCE + CEF)
(HJGB = HBJK + BGK)
(HJCE + EEF + BGK + ABKJ =) (BGHECJKB + ABC + ABKJ)
(BGHEC + ABC + ABJK + JKC)

(BEDC = BC^2)
(BGE = DCF)
(EHD = BAC)
(BKCDHG = BKCDHG BGK = JKC)
therefore
(BKCDHG + ABC + DFC =)
(KKCDHG + BGE + EHD)

(EB = BC), (ED = BC), (CD = BC)
(ABC), (BGE), (EHD), (DFC) are right triangles on equal hypotenuses, and therefore equal, since all the angles correspond; therefore, triangle (EGB = DCF) and (EHD = ABC); [ illegible] (BGHD:CK) is equal to itself.