Lemma 28

There is no oval figure whose area, cut off by right lines at pleasure, can be universally found by means of equations of any number of finite terms and dimensions.

A right line as a pole is perpetually revolving with an uniform motion in an oval.

In that right line a moveable point going out from the pole moves always forward with a speed proportional to the square of that right line within the oval.

By this motion that point will describe a spiral with infinite circumgyrations.

If a portion of the area of the oval cut off by that right line could be found by a finite equation, the distance of the point from the pole, which is proportional to this area, might be found by the same equation.

Therefore all the points of the spiral might be found by a finite equation

Therefore the intersection of a right line given in position with the spiral might also be found by a finite equation.

But every right line infinitely produced cuts a spiral in an infinite number of points.

The equation by which any one intersection of two lines is found at the same time exhibits all their intersections by as many roots, and therefore rises to as many dimensions as there are intersections. Be cause two circles mutually cut one another in two points, one of those intersections is not to be found but by an equation of two dimensions, by which the other intersection may be also found.

There may be 4 intersections of two conic sections, any one of them is not to be found universally, but by an equation of four dimensions, by which they may be all found together.

For if those intersections are severally sought, because the law and condition of all is the same, the calculus will be the same in every case, and therefore the conclusion always the same; which must therefore comprehend all those intersections at once within itself, and exhibit them all indifferently.

Hence it is that the intersections of the conic scions with the curves of the third order, because they may amount to six, come out together by equations of 6 dimensions;

The intersections of two curves of the third order, because they may amount to nine, come out together by equations of nine dimensions.

If this did not necessarily happen, we might reduce all solid to plane Problems, and those higher than solid to solid Problems.

These curves are irreducible in power.

If the equation by which the curve is defined may be reduced to a lower power, the curve will not be one single curve, but composed of two, or more, whose intersections may be severally found by different calculusses.

After the same manner the 2 intersections of right lines with the conic sections come out always by equations of two dimensions;

The 3 intersections of right lines with the irreducible curves of the third order by equations of 3 dimensions;

The 4 intersections of right lines with the irreducible curves of the fourth order, by equations of four dimensions; and so on in infinitum.

Wherefore the innumerable intersections of a right line with a spiral, since this is but one simple curve and not reducible to more curves, require equations infinite in number of dimensions and roots, by which they may be all exhibited together.

For the law and calculus of all is the same.

For if a perpendicular is let fall from the pole upon that intersecting right line, and that perpendicular together with the intersecting line revolves about the pole, the intersections of the spiral will mutually pass the one into the other; and that which was first or nearest, after one revolution, will be the second;

After two, the third; and so on: nor will the equation in the mean time be changed but as the magnitudes of those quantities are changed, by which the position of the intersecting line is determined. Wherefore since those quantities after every revolution return to their first magnitudes, the equation will return to its first form;

Consequently, one and the same equation will exhibit all the intersections, and will therefore have an infinite number of roots, by which they may be all exhibited. And therefore the intersection of a right line with a spiral cannot be universally found by any finite equation; and of consequence there is no oval figure whose area, cut off by right lines at pleasure, can be universally exhibited by any such equation.

By the same argument, if the interval of the pole and point by which the spiral is described is taken proportional to that part of the perimeter of the oval which is cut off; it may be proved that the length of the perimeter cannot be universally exhibited by any finite equation. But here I speak of ovals that are not touched by conjugate figures running out in infinitum.

Cor.

Hence the area of an ellipsis, described by a radius drawn from the focus to the moving body, is not to be found from the time given by a finite equation; and therefore cannot be determined by the description of curves geometrically rational. Those curves I call geometrically rational, all the points whereof may be determined by lengths that are definable by equations; that is, by the complicated ratios of lengths.

Other curves (such as spirals, quadratrixes, and cycloids) I call geometrically irrational. For the lengths which are or are not as number to number (according to the tenth Book of Elements) are arithmetically rational or irrational. And therefore I cut off an area of an ellipsis proportional to the time in which it is described by a curve geometrically irrational, in the following manner.