I am not able to let this occasion pass without speaking to you in regard to certain of my meditations since I had the honor of seeing you. Among other things I have made quite a number of investigations into jurisprudence and it seems to me that something permanent and useful might be established, quite as much for the sake of having ascertained laws, of which there is a great lack in Germany and perhaps also in France, as also for the establishment of short and good forms of procedure. For this purpose it is not sufficient to be strict with regard to the terms or the established days and other conditions, as is the case with the laws compiled under the code of Louis; for to suffer a good cause to be lost because of formalities, is in jurisprudence a remedy comparable to that of a surgeon who is continually cutting off arms and legs. They say that the King is having work done for the reform of chicanery, and I think that something of importance might be done along this line.

I have also been interested in the subject of mines, because of those which we have in our country; and I have frequently visited them by command of the Prince. I think I have made several discoveries in regard to the formation, not so much of the metals as of those forms in which the metals are found and of certain bodies among which they lie. For example, I have shown the manner of the formation of slate.

Besides this I have gathered together memoirs and titles concerning the history of Brunswick, and recently I read a document regarding the boundaries of the Hildesheim bishopric of the canonized Emperor Henry II., where I was surprised to find these words, "for the safety of his royal wife and child." This seemed to me to be quite contrary to the accepted opinion which would have us believe that he maintained a state of virginity toward his wife, St. Cunigunde.

Besides this I have diverted myself frequently with abstract thoughts in metaphysics and geometry. I have discovered a new method of tangents, which I have had printed in the Journal of Leipsic. You know, that Hudde and later De Sluse developed this matter quite far, but there were two things lacking. The one was that when the unknown term or indeterminate was expressed in fractions and irrationals, these had to be eliminated in order to use their methods, which made the calculation assume an extent and an elaborateness very awkward and often unmanageable; while my method is not encumbered at all with fractions or irrationals. This is why the English have made so much of it. The other fault of the method of tangents is that it does not apply to the lines which Descartes calls mechanical and which I prefer to call transcendental; while my method applies to them just the same, and I can calculate the tangent of the cycloid or of any other line. I claim also to give in general the means of reducing these lines to calculation, and I hold that they must be received into geometry, whatever M. Descartes may say. My reason is that there are analytical problems which are of no degree or whose degree is required; e.g., to cut an angle in the incommensurable ratio of one straight line to another straight line. This problem is neither in plane geometry nor in solid nor in super-solid geometry, it is, nevertheless, a problem, and for this reason I call it transcendental. Such is also this problem: Solve the following equation: X(x) + X = 30, where the unknown term X is found also in the exponent and the degree also of an equation is required. It is easy to find here that X is equal to 3 for 3(3) + 3 or 27 + 3 makes 30. But it is not always so easy to solve it, above all when the exponent is not a rational number; and we must have recourse to lines or loci which are appropriate to the purpose and which therefore must be admitted into geometry. Now I show that the lines which Descartes would exclude from geometry depend upon equations which transcend algebraic degrees but are yet not beyond analysis, nor geometry. I therefore call the lines, which M. Descartes accepts, algebraic because they are of a certain degree in an algebraic equation. The others I call

transcendental. These I reduce to calculations, and their construction I show either through points or through motion; and, if I might venture to say, I claim to advance analysis thereby

ultra Herculis columnas.