Newton's Constructions

The 5-Point Conic Section Construction

Through five given points, construct the conic section. The proof justifying this construction appears as Principia, Book 1, Proposition 22, which refers back to Lemmas 17 and 18, which in turn depend on Apollonius, Conica, Book III, Propositions 17, 19, 21, and 23. These are recommended reading, and are not really as daunting as they may appear here. It was the habit of Apollonius to present the same concept in several separate propositions with minor variations in the conditions, which is why four of his propositions are listed above. Newton’s Lemmas 17 and 18 are actually converses of each other.

Let the given points be A, B, C, D, and P. Construct quadrilateral ABCD with the sides produced as lines. Here it does not need to be a quadrilateral in the strictest sense, which is to say, it does not matter where else these lines intersect, as long as no three of the given points are collinear. It could be twisted into an anti-quadrilateral without interfering with the construction. Points B and D are what Newton called the poles. Through P construct line j_AB parallel to AB, and j_AD parallel to AD. Let j_AB meet BC at Q, and let j_AD meet CD at R. Draw line QR. The direction of this line will become important.

Let k be a line drawn in an arbitrary direction through D. The objective now is to construct the other point where line k meets the section. Let lines k and j_AD intersect at F. Through F construct the line parallel to QR, and let it intersect j_AB at G. Lines BG and k meet at point H, which lies on the section.

Newton, of course, had no dynamic geometry software, but it seems reasonable to suppose he imagined something like that. After constructing this single point, he considered the whole section effectively constructed. That is because line k was defined by point D alone. It can lie in any direction, which means it can lie in all directions.

Here I have translated point D 100 pixels in the 12 o’clock direction in order to define the center of a circle through D. Having this measured distance and direction may seem to violate construction rules, but it is still in the spirit of the game. The circle only serves as a path for a point driving the construction of a locus. Let L be a point having the circle as its path, and let line k now be defined as DL. As L travels its path, line k rotates through 180°, taking it through all possible directions, so the locus of H is all points on the section. The locus is the section.

Tangent Line Construction

Given five points on a conic section, construct the tangent line through one of those points. Newton solves this in Principia, Book 1, Lemma 20, Corollary 1. We can continue with the section that was just constructed. Line BH is a secant line of the section. As H approaches B (more evidence that Newton imagined dynamic geometry with moving points), HB approaches tangency. Suppose H is moved to coincide with B. In that case, line k is actually DB. Walk through the construction again. Lines k and j_AD meet at F. From F a line parallel to QR is drawn, meeting j_AB at G. And GB intersects k at H, which, as we have defined it, is simply B itself. Line BG is tangent to the section at B.

Since k goes through both poles, it is possible to work from the other end, B, to find another tangent line. Let lines k and j_AB meet at U. From there a line parallel to QR meets j_AD at V. Line DV is tangent to the section at D.

The completed construction should have all of the related parts (axial vertices, axes, foci, directrices, asymptotes). None of those objects are given in the general challenges, which complicates matters. This is handled by the Conic Parameters construction, which requires that we begin with tangent lines through three points on the curve. This construction so far has only two tangent lines, through the poles B and D. To find tangents through other points, define poles other than B and D, and start again.

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