French Transcription
In transcribing Clairaut’s text (pdf), we remained faithful to his sentence structure, grammar, and word choice. We did, however, modernize antiquated spellings and add accenting as appropriate for modern French. We also expanded abbreviations. The original spelling, accenting, or abbreviations are given in critical apparatus, which appear as a collection of footnotes on the French pages. The lines of the text are numbered for reference, but the reader should note that line numbers do not appear in the original text and do not correspond in any way to the lines of the original text.
For the mathematical content of the French transcription, we maintained the notation Clairaut chose as much as possible. For example, Clairaut often has expressions such as
√[\(xx + yy\)],
and we have preserved these as much as is possible with \(\LaTeX\).
The critical apparatus for the French consists of three groupings:

Group A: Textual. These footnotes indicate textual changes, such as a modernization of spelling or accenting. For example, the word “côtez” appears in the original text, but we have updated it to read “côtes”. The verb côtes appears in the main text, and the corresponding footnote reads 2 côtes ] côtez. The number corresponds to the line number on which the word occurs; the word preceding the bracket indicates the spelling in our edition; the word after the bracket indicates the spelling in the original.

Group B: Reading. These notes are intended to clarify the meaning of a word in French. For example, Clairaut uses the word “quarrant”. This is, however, an antiquated word; a modern equivalent would be the word “carrer” (“to square”).

Group C: Mathematical. These notes indicate corrections to the mathematical content. For example, Clairaut’s original text has \(x^{n+1}\) in a place where it should be \(x^{n+2}\). These notes also indicate issues with the diagrams, such as an omitted label in Clairaut’s original.
English Translation
We chose to not attempt a literal, wordforword translation (pdf), and have instead taken some small liberties to give one with clearer meaning. For example, Clairaut employed a plethora of semicolons in his text. If we interpret them as periods, the result is a string of sentence fragments. If we interpret them as semicolons, it results in a massive runon sentence. Accordingly, we have been flexible with sentence breaks, and have done so both to make the text readable and to remain true to what we think was Clairaut’s intent.
For mathematical notation in the English, we have used modern styling. For example, we replace Clairaut’s \(xx\) with \(x^2\) , and we render surds as square roots, e.g. \(\sqrt{x^2 + y^2}\) instead of √\(xx + yy\).
Notes in the English translation are of two varieties:
 Translation. These items comment on specific aspects of our translation.
 Mathematical. These items explicate mathematical details, such as intermediate steps not justified by Clairaut.
General Editorial Conventions
In both the English and French, we have taken the liberty of placing most of the mathematical notation on its own line. The source text prints them inline, and this is often detrimental to the readability of complicated notation. Equations count as a new line for the purposes of line numbering, and each equation is numbered in the usual \(\LaTeX\) style.
The mathematical expressions are formatted in two ways. Points are given in “math bold” style, e.g. the point \(\mathbf{A}\) or the point \(\mathbf{n}\). Algebraic quantities are given in standard \(\LaTeX\) italics, e.g. \(x, y, a, m, n\). This was necessary in order to distinguish the points \(\mathbf{a}\) and \(\mathbf{n}\) from the algebraic quantities \(a\) and \(n\), as Clairaut used both throughout the paper. This is also advantageous because it distinguishes between the point \(\mathbf{a}\), the algebraic quantity \(a\), and the common French verb “a”, a conjugation of “avoir”.
The figures were created using GeoGebra, enabling the reader to manipulate the positions of some points. Clairaut’s curves are given in color (red or blue), and other lines are given in black. Some lines are dashed: often this is because Clairaut did so in his diagrams, but on a few occasions, we have made lines dashed so as not to distract from the more important parts of the figures. The coordinate system requires explanation: Clairaut oriented the \(x\) axis running vertically, and \(y\) axis running horizontally. In order to graph Clairaut’s curves, it was necessary to switch \(x\) and \(y\), since GeoGebra requires the modern standard coordinate system of \(x\) running horizontally and \(y\) vertically. We have not changed \(x\) and \(y\) in the text. If the reader desires to graph the curves themselves, e.g. with Mathematica or GeoGebra, it will be necessary to switch \(x\) and \(y\) as they appear in the text so that the graphs match the orientation used by Clairaut.