Synopsis of Universal Geometry

ON THE SYNOPSIS

A magnitude is greater than a given magnitude by a set amount when, after that set amount is subtracted, the remainder is equal to it. This refers to Euclid’s Data, Proposition 12.

A magnitude is greater than a given magnitude in ratio when, after a set amount is subtracted, the remainder has a given ratio to that same amount. This refers to Euclid’s Data, Propositions 10, 11, and 13 through 21.

Although a very learned man and most dear friend noticed that propositions 48, 49, 74, 75, 79, 82, 83, and 89 were not fully included, we have now provided those data. Furthermore, we have presented those things which have been restored thus far concerning the Section of Space original: "Spatij Sectione," referring to a lost work by Apollonius of Perga, the Determinate Section original: "determinata Sectione," another lost Apollonius text, Tangencies original: "Tactionibus," regarding the problem of drawing circles tangent to given points, lines, or circles both plane and spherical, Plane Loci, and Inclinations. Of these, Euclid’s Porisms original: "Euclidis Porismata," a famous lost work on the properties of geometric curves still remain to be restored. As for the problem of the "locus relative to three or four lines," our Geometer This likely refers to Pierre de Fermat or Girard Desargues, who were working on these classical problems at the time. has just now demonstrated it.

Since the majority of the propositions of Pappus Pappus of Alexandria, a 4th-century mathematician whose "Collection" preserved much of Greek geometry. require diagrams, I have only included those that seem capable of being understood without figures. These may perhaps be given in another edition with figures placed in the margins of the book.

Furthermore, anyone who wishes to see the Greek text of Pappus should know it is held by Lord Lescuyer, a distinguished Senator, whose library is second to none. It will be helpful to mention here that problem which I presented at the end of the Preface to the Conics by the famous Mydorge Claude Mydorge, a French mathematician and friend of Descartes.. It is now proposed more generally by G. Desargues Girard Desargues, the founder of projective geometry. to be solved by the learned as follows:

Given a solid, as discussed in that same place; find the position of a cutting plane that intersects it in a figure of a specific type, where the axes of the figure are in a set ratio; or where the maximum inclination of the conjugate diameters equals a given inclination.

To solve this, by means of two lines described through any number of points, he finds the position of the plane cutting the solid in an elliptical figure, from whose center a straight line drawn to the vertex of the solid is perpendicular to the plane. By means of this elliptical figure, the position of a plane cutting the solid in a circle will be found. With the help of this circle, one can find the position of a plane cutting the solid in a figure of a given type having its axes in a set ratio; or whose conjugate axes of maximum inclination are equal to the given inclination. But perhaps these same things can be found without these means, starting from the given primary base of the aforementioned solid.

There follows his most general proposition, which the Geometer may solve: Given the base and vertex of a cone, the intersection of the plane of the base with the plane cutting the cone, and the inclination The angle at which the two planes meet. of these two planes, find—without drawing a figure—the planes cutting the conic sections that generate diameters at a given angle, tangents, and ordinates.