Abstract
A short early mathematical piece (1639) on conic sections, with definitions and lemmas indebted to Desargues that lead to the famous mystic-hexagram theorem. A purely geometric text with no philosophical content.
Traduzione di Frances Marguerite Clarke · Delphi Classics, 2000 · pubblico dominio
First Definition
When several straight lines meet at the same point, or are parallel to each other, all these lines are said to be of the same order or of the same ordonnance, and the totality of these lines is termed an order of lines, or an ordonnance of lines. [This definition is taken almost word for word from DESARGUES. See the notes to the BRUNSCHVICG and BOUTROUX edition, t. I., Paris, 1908. This translation is made from the facsimile of the original as given in this edition, and acknowledgment is hereby made of the assistance rendered by these notes in determining the meaning of several passages. It should also be said that the text of this edition is in marked contrast to the imperfect one given in the Paris edition of 1819.]
Definition II
By the expression “conic section,” we mean the circle, ellipse, hyperbola, parabola, and an angle; since a cone cut parallel to its base, or through its vertex, or in the three other directions which produce respectively an ellipse, a hyperbola, and a parabola, produces in the conic surface, either the circumference of a circle, or an angle, or an ellipse, a hyperbola, or a parabola.
Definition III
By the word “droite” (straight) used alone, we mean “ligne droite” (straight line).
[In this translation, the word “line,” meaning a straight line-segment, will be used for “droite.”]
Lemma I
If in the plane M, S, Q, two straight lines MK, MV, are drawn from point M and two lines SK, SV from point S; and if K be the point of intersection of the lines MK, SK; V, the point of intersection of the lines MV, SV; A, the point of intersection of the lines MA, SA; and μ, the point of intersection of the lines MV, SK; and if through two of the four points A, K, μ, V, which can not lie in the same line with points M, S, and also through points K, V, a circle passes cutting the lines MV, MP, SV, SK at points O, P, Q, N, then I say that the lines MS, NO, PQ, are of the same order.
Lemma II
If through the same line several planes are passed, and are cut by another plane, all lines of intersection of these planes are of the same order as the line through which these planes pass.
On the basis of these two lemmas and several easy deductions from them, we can demonstrate that if the same things are granted as for the first lemma, that is, through points K, V, any conic section whatever passes cutting the lines MK, MV, SK, SV in points P, O, N, Q, then the lines MS, NO, PQ will be of the same order. This constitutes a third lemma. [This involves the so-called “Mystic Hexagram,” the dual of Brianchon’s Theorem. Pascal did not state the hexagram theorem in the form commonly seen in textbooks.]
By means of these three lemmas and certain deductions, therefrom, we propose to derive a complete ordered sequence of conics, [.. des éléments coniques complets] that is to say, all the properties of diameters and other straight lines, [… et côtés droits] of tangents, &c., the construction of the cone from substantially these data, the description of conic sections by points, etc.
Having done this, we shall state the properties which follow, doing this in a more general manner than usual. Take for example, the following: If in the plane MSQ, in the conic PKV, there are drawn the lines AK, AV, cutting the conic in points P, K, Q, V; and if from two of these four points, which do not lie in the same line with point A, — say the points K, V, and through two points N, O, taken on the conic, there are produced four lines KN, KO, VN, VO, cutting the lines AV, AP at points L, M, T, S, — then I maintain that the proportion composed of the ratios of the line PM to the line MA, and of the line AS to the line SQ, is the same as the proportion composed of the ratio of the line PL to the line LA, and of the line AT to the line TQ.
We can also demonstrate that if there are three lines DE, DG, DH that are cut by the lines AP, AR at points F, G, H, C, y, B, and if the point E be fixed in the line DC, the proportion composed of the ratios of the rectangle EF.FG to the rectangle EC.Cy, and of the line Ay to the line AG, is the same as the ratio of the rectangle EF.FH to the rectangle EC.CB, and of the line AB to the line AH. The same is also true with respect to the ratio of the rectangle FE.FD to the rectangle CE.CD. Consequently, if a conic section passes through the points E, D, cutting the lines AH, AB in points P, K, R, ψ, the proportion composed of the ratio of the rectangle of these lines EF, FC, to the rectangle of the lines EC, Cγ, and of the line γA to the line AG, will be the same as the ratio of the rectangle of the lines FK, FP, to the rectangle of the lines CR, Cψ, and of the rectangle of the lines AR, Αψ, to the rectangle of the lines AK, AP.
We can also show that if four lines AC, AF, EH, EL intersect in points N, P, M, O, and if a conic section cuts these lines in points C, B, F, D, H, G, L, K, the proportion consisting of the ratios of rectangle MC.MB to rectangle PF.PD, and of rectangle AD.AF to rectangle AB.AC, is the same as the proportion composed of the ratios of rectangle ML.MK to the rectangle PH.PG, and of rectangle EH.EG to rectangle EK.EL.
We can also demonstrate a property stated below, due to M. DESARGUES of Lyons, one of the great geniuses of this time and well versed in mathematics, particularly in conics, whose writings on this subject although few in number give abundant proof of his knowledge to those who seek for information. I should like to say that I owe the little that I have found on this subject to his writings, and that I have tried to imitate his method, as far as possible, in which he has treated the subject without making use of the triangle through the axis.
Giving a general treatment of conic sections, the following is the remarkable property under discussion: If in the plane MSQ there is a conic section PQN, on which are taken four points K, N, O, V from which are drawn the lines KN, KO, VN, VO, in such a way that through the same four points only two lines may pass, and if another line cuts the conic at points R, ψ, and the lines KN, KO, VN, VO, in points X, Y, Z, δ, then as the rectangle ZR.Zψ is to the rectangle γR.γψ, so the rectangle δR.δφ is to the rectangle XR.Xψ.
We can also prove that, if in the plane of the hyperbola, the ellipse, or the circle AGE of which the center is C, the line AB is drawn touching the section at A, and if having drawn the diameter we take line AB such that its square shall be equal to the square of the figure, [In order that the square of segment AB, which is equal to DE + DF, shall be equal to one fourth of the circumscribed rectangle, the conic must be a circle. If the conic is an ellipse, AB will be taken equal to the axis which is perpendicular to CA. DESARGUES treated analogous questions in his Brouillon Projet (Œuvres de DESARGUES, I, p. 202 et p. 284).] and if CB is drawn, then any line such as DE, parallel to line AB and cutting the section in E, and the lines AC, CB in points D, F, then if the section AGE is an ellipse or a circle, the sum of the squares of the lines DE, DF will be equal to the square of the line AB; and in the hyperbola, the difference between the same squares of the lines DE, DF will be equal to the square of the line AB.
We can also deduce [from this] several problems; for example:
From a given point to draw a tangent to a given conic section.
To find two diameters that meet in a given angle.
To find two diameters cutting at a given angle and having a given ratio.
There are many other problems and theorems, and many deductions which can be made from what has been stated above, but the distrust which I have, due to my little experience and capacity, does not allow me to go further into the subject until it has passed the examination of able men who may be willing to take this trouble. After that if someone thinks the subject worth continuing, I shall endeavor to extend it as far as God gives me the strength.
At Paris, M.DC.XL.