therefore the safer

engine for their purposes.

With the lawyers

it is a new thing. They have, in the mother country, been generally the firmest supporters of the free principles of their constitution. But there too they have changed. I ascribe much of this to the substitution of Blackstone for my Lord Coke, as an elementary work. In truth, Blackstone and Hume have made tories of all England, and are making tories of those young Americans whose native feelings of independence do not place them above the wily sophistries of a Hume or a Blackstone. These two books, but especially the former, have done more towards the suppression of the liberties of man, than all the million of men in arms of Bonaparte and the millions of human lives with the sacrifice of which he will stand loaded before the judgment seat of his Maker. I fear nothing for our liberty from the assaults of force; but I have seen and felt much, and fear more from English books, English prejudices, English manners, and the apes, the dupes, and designs among our professional crafts. When I look around me for security against these seductions, I find it in the wide-spread of our agricultural citizens, in their unsophisticated minds, their independence and their power, if called on, to crush the Humists of our cities, and to maintain the principles which severed us from England. I see our safety in the extent of our confederacy, and in the probability that in the proportion of that the sound parts will always be sufficient to crush local poisons. In this hope I rest, and tender you the assurance of my esteem and respect.

TO MR. GIRARDIN.

MONTICELLO, March 18, 1814.

DEAR SIR,-According to your request of the other day, I send you my formula and explanation of Lord Napier's theorem, for the solution of right-angled spherical triangles. With you I think it strange that the French mathematicians have not used or noticed this method more than they have done. Montucla, in

his account of Lord Napier's inventions, expresses a like surprise at this fact, and does justice to the ingenuity, the elegance, and convenience of the theorem, which, by a single rule easily preserved in the memory, supplies the whole table of cases given in the books of spherical trigonometry. Yet he does not state the rule, but refers for it to Wolf, Cours de Mathematiques. I have not the larger work of Wolf; and in the French translation of his abridgement, (by some member of the congregation of St. Maur,) the branch of spherical trigonometry is entirely omitted. Potter, one of the English authors of Courses of Mathematics, has given the Catholic proposition, as it is called, but in terms unintelligible, and leading to error, until, by repeated trials, we have ascertained the meaning of some of his equivocal expressions. In Robert Simson's Euclid we have the theorem with its demonstrations, but less aptly for the memory, divided into two rules, and these are extended as the original was, only to the cases of right-angled triangles. Hutton, in his Course of Mathematics, declines giving the rules, as "too artificial to be applied by young computists." But I do not think this. It is true that when we use them, their demonstration is not always present to the mind; but neither is this the case generally in using mathematical theorems, or in the various steps of an algebraical process. We act on them, however, mechanically, and with confidence, as truths of which we have heretofore been satisfied by demonstration, although we do not at the moment retrace the processes which establish them. Hutton, however, in his Mathematical Dictionary, under the terms "circular parts," and "extremes," has given us the rules, and in all their extensions to oblique spherical, and to plane triangles. I have endeavored to reduce them to a form best adapted to my own frail memory, by couching them in the fewest words possible, and such as cannot, I think, mislead, or be misunderstood. My formula, with the explanation which may be necessary for your pupils, is as follows:

Lord Napier noted first the parts, or elements of a triangle, to wit, the sides and angles; and expunging from these the right

angle, as if it were a non-existence, he considered the other five parts, to wit, the three sides, and two oblique angles, as arranged in a circle, and therefore called them the circular parts; but chose, (for simplifying the result,) instead of the hypothenuse and two oblique angles, themselves, to substitute their complements. So that his five circular parts are the two legs themselves, and the complements of the hypothenuse and of the two oblique angles. If the three of these, given and required, were all adjacent, he called it the case of conjunct parts, the middle element the MIDDLE PART, and the two others the EXTREMES disjunct from the middle or EXTREMES DISJUNCT. He then laid down his catholic rule, to wit:

"The rectangle of the radius, and sine of the middle part, is equal to the rectangle of the tangents of the two EXTREMES CONJUNCT, and to that of the cosines of the two EXTREMES DISJUNCT."

And to aid our recollection in which case the tangents, and in which the cosines are to be used, preserving the original designations of the inventor, we may observe that the tangent belongs to the conjunct case, terms of sufficient affinity to be associated in the memory; and the sine complement remains of course for the disjunct case; and further, if you please, that the initials. of radius and sine, which are to be used together, are alphabetical consecutives.

Lord Napier's rule may also be used for the solution of oblique spherical triangles. For this purpose a perpendicular must be let fall from an angle of the given triangle internally on the base, forming it into two right-angled triangles, one of which may contain two of the data. Or, if this cannot be done, then letting it fall externally on the prolongation of the base, so as to form a right-angled triangle comprehending the oblique one, wherein two of the data will be common to both. To secure two of the data from mutilation, this perpendicular must always be let fall from the end of a given side, and opposite to a given angle.

But there will remain yet two cases wherein Lord Napier's rule cannot be used, to wit, where all the sides, or all the angles

alone are given. To meet these two cases, Lord Buchan and Dr. Minto devised an analogous rule. They considered the sides themselves, and the supplements of the angles as circular parts in these cases; and, dropping a perpendicular from any angle. from which it would fall internally on the opposite side, they assumed that angle, or that side, as the MIDDLE part, and the other angles, or other sides, as the OPPOSITE OF EXTREME parts, disjunct in both cases. Then "the rectangle under the tangents of half the sum, and half the difference of the segments of the MIDDLE part, is equal to the rectangle under the tangents of half the sums, and half the difference of the OPPOSITE PARTS."

And, since every plane triangle may be considered as described on the surface of a sphere of an infinite radius, these two rules may be applied to plane right-angled triangles, and through them to the oblique. But as Lord Napier's rule gives a direct solution only in the case of two sides, and an uncomprised angle, one, two, or three operations, with this combination of parts, may be necessary to get at that required.

You likewise requested for the use of your school, an explanation of a method of platting the courses of a survey, which I mentioned to you as of my own practice. This is so obvious and simple, that as it occurred to myself, so I presume it has to others, although I have not seen it stated in any of the books. For drawing parallel lines, I use the triangular rule, the hypothe

nusal side of which being applied to the side of a common straight rule, the triangle slides on that, as thus, always parallel to itself. Instead of drawing meridians on his paper, let the pupil draw a parallel of latitude, or east and west line, and note in that a point for his first station, then applying to it his protractor, lay off the first course and distance in the usual way to ascertain his second station. For the second course, lay the triangular rule to the east and west line, or first parallel, holding the straight or guide rule firmly against its hypothenusal side. Then slide up the triangle (for a northerly course) to the point of his

second station, and pressing it firmly there, lay the protractor to that, and mark off the second course, and distance as before, for the third station. Then lay the triangle to the first parallel again, and sliding it as before to the point of the third station, there apply to it the protractor for the third course and distance, which gives the fourth station; and so on. Where a course is southwardly, lay the protractor, as before, to the northern edge of the triangle, but prick its reversed course, which reversed again in drawing, gives the true course. When the station has got so far from the first parallel, as to be out of the reach of the parallel rule sliding on its hypothenuse, another parallel must be drawn by laying the edge, or longer leg of the triangle to the first parallel as before, applying the guide-rule to the end, or short leg, (instead of the hypothenuse,) as in the margin, and sliding the triangle up to the point for the new parallel. I have found this, in practice, the quickest and most correct method of platting which I have ever tried, and the neatest also, because it disfigures the paper with the fewest unnecessary lines.

If these mathematical trifles can give any facilities to your pupils, they may in their hands become matters of use, as in mine they have been of amusement only.

Ever and respectfully yours.

TO M. DUFIEF.

MONTICELLO, April 19, 1814.

DEAR SIR,-Your favor of the 6th instant is just received, and I shall with equal willingness and truth, state the degree of agency you had, respecting the copy of M. de Becourt's book, which came to my hands. That gentleman informed me, by letter, that he was about to publish a volume in French, "Sur la Création du Monde, un Systême d'Organisation Primitive," which, its title promised to be, either a geological or astronomical work. I subscribed; and, when published, he sent me a copy;