“Let the letters a b c denote the three angular points of a rectilineal triangle. If the point did move continuously over the lines ab, bc, ca, that is, over the perimeter of the figure, it would be necessary for it to move at the point b in the direction ab, and also at the same point b in the direction bc. These motions being diverse, they cannot be simultaneous. There-fore, the moment of presence of the movable point at vertex b, considered as moving in the direction ab, is different from the moment of presence of the movable point at the same vertex b, considered as moving in the same direction bc. But between two moments there is time; therefore, the movable point is present at point b for some time, that is, it rests. Therefore it does not move continuously, which is contrary to the assumption. The same demonstration is valid for motion over any right lines including an assignable angle. Hence a body does not change its direction in continuous motion except by following a line no part of which is straight, that is, a curve, as Leibnitz maintained.”

— Immanuel Kant

Kant's Inaugural Dissertation (1770), Section III On The Principles Of The Form Of The Sensible World

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Immanuel Kant 200

German philosopher 1724–1804

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“The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line.
This then was the first step which led to a new branch of mathematics… I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea.”

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“How could one argue with a man who was always drawing lines and circles to explain the position; who, one day, drew a diagram [here Michael illustrated with pen and paper] saying 'take a point A, draw a straight line to point B, now three-fourths of the way up the line take a point C. The straight line AB is the road to the Republic; C is where we have got to along the road, we canot move any further along the straight road to our goal B; take a point out there, D [off the line AB]. Now if we bend the line a bit from C to D then we can bend it a little further, to another point E and if we can bend it to CE that will get us around Cathal Brugha which is what we want!”

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“The discovery of Hippocrates amounted to the discovery of the fact that from the relation
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The first solution now consists in drawing a parabola, with vertex O and axis Ox, such that its parameter is equal to BO or b, and a hyperbola with Ox, Oy as asymptotes such that the rectangle under the distances of any point on the curve from Ox, Oy respectively is equal to the rectangle under AO, BO i. e. to ab. If P be the point of intersection of the parabola and hyperbola, and PN, PM be drawn perpendicular to Ox, Oy, i. e. if PN, PM be denoted by y, x, the coordinates of the point P, we shall have

\begin{cases}y^2 = b. ON = b. PM = bx\\ and\\ xy = PM. PN = ab\end{cases}whence\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.
In the second solution of Menaechmus we are to draw the parabola described in the first solution and also the parabola whose vertex is O, axis Oy and parameter equal to a.”

Thomas Little Heath (1861–1940) British civil servant and academic

The point P where the two parabolas intersect is given by<center><math>\begin{cases}y^2 = bx\\x^2 = ay\end{cases}</math></center>whence, as before,<center><math>\frac{a}{x} = \frac{x}{y} = \frac{y}{b}.</math></center>
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“As for the moral status of majority rule, it must be pointed out that it allows for A and B to band together to rip off C, C and A in turn joining to rip off B, and then B and C conspiring against A, and so on.”

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(3) Whenever a point P exists which is incident on both the circles A(B), C(D), that point is regarded as determinate.
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“I guess I just don't get the point. It's like, why should you bother getting attached to anything if,

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Alyson Nöel (1965) writer

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Immanuel Kant 200

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