Geometry on Superphysics

Problems that can be constructed using only circles and straight lines

Mon, 01 Jan 0001 00:00:00 +0000

This treatise can only be understood by those who are familiar with Geometry.

All Geometry problems can be easily reduced to such terms that it is only necessary to know the lengths of certain straight lines to construct them.

Geometry Simplified

Mon, 01 Jan 0001 00:00:00 +0000

The method of expressing the value of all the roots of cubic equations: and subsequently of all those that rise only to the biquadratic.

For the rest, this way of expressing the value of the roots by the ratio they have to the sides of certain cubes of which only the volume is known, is as simple as expressing them by the ratio they have to the chords of certain arcs, or portions of circles, whose triple is given.

Pappus

Mon, 01 Jan 0001 00:00:00 +0000

Thus, when one wants to solve a problem, one must first consider it as already completed, and assign names to all the lines that seem necessary for its construction—both those that are unknown and the others.

The Nature of Curved Lines

Mon, 01 Jan 0001 00:00:00 +0000

The ancients rightly observed that:

among geometrical problems, some are planar, others solid, and others linear
some can be constructed using only straight lines and circles
others require, at a minimum, the use of conic sections
others require the use of more complex curves

But I am surprised that they:

Problems that can be constructed using only circles and straight lines

Mon, 01 Jan 0001 00:00:00 +0000

Consider the lines AB, AD, AF, and similar ones traced with the instrument YZ.

The instrument is composed of several rulers joined together so that YZ is along line AN.

The Construction of Problems That Are Solid or More Than Solid

Mon, 01 Jan 0001 00:00:00 +0000

All curved lines that can be described by some regular motion should be accepted in geometry.

How We Can Reduce The Number Of Dimensions Of An Equation

Mon, 01 Jan 0001 00:00:00 +0000

How one can reduce the number of dimensions of an equation when one knows one of its roots and How one can examine whether a given quantity is the value of a root

The sum of an equation which contains several roots can always be divided by a binomial composed of the unknown quantity minus the value of one of the true roots—whichever it may be—or plus the value of one of the false roots; by this means, its degree is reduced accordingly.

How to Solve using an equation with up to 6 Dimensions

Mon, 01 Jan 0001 00:00:00 +0000

General method for constructing all problems reduced to an equation that has no more than six dimensions.

You already know how, when seeking the quantities required for the construction of these problems, they can always be reduced to some equation, which only goes up to the square of a cube, or, on p. 403, to the sursolid.

Geometry Simplified

Mon, 01 Jan 0001 00:00:00 +0000

If the method of tracing the line ACN by the movement of a parabola seems inconvenient to you, it is easy to find several other ways to describe it.

For example, if having the same quantities as before for AB and BL and the same for BK, which was taken as the principal latus rectum of the parabola; we describe the semicircle KST whose center is taken arbitrarily on the line BK, such that it intersects the line AB somewhere, as at point S, and that from point T, where it ends, we take towards K the line TV, equal to BL; then having drawn the line SV, we draw another parallel to it through point A, such as AC;