sive field of speculative science, not much frequented even by mathematicians. I would only observe, that geometricians have, long since, demonstrated, generally, the impossibility of solving geometrical problems of the second degree, or order, by any lines of the first order, since these cannot be so combined, as to involve the more complicated conditions and relations, necessarily implied in such problems.

The other problem, which is the subject of Mr. Winthrop's second paper, is that of the trisection of an angle, to which though equally capable of a solution by right lines, as that of the duplication of the cube, a dissimilar one has been given from the consideration of lines of a superior order, in a manner consonant to the strict principles of geometrical constructions, and which appears to be not less novel than ingenious. The author, however, has omitted to investigate the nature and specific properties of the curve, called by him, the trisecting curve; but it is easily shown, that it is no other than the common hyperbola: For, using his scheme, suppose VC,* the curve, AB the directrix, V the vertex, P any point in the curve; from P, draw Pd perpendicular to FV; and VA, PI perpendicular to the directrix AB; draw the line FP from the focus F to the point P; put VA=a, Vd=x, then PI=a+x, FP=2 a+2x, and Fd=2a-x; let Pd-y. Now in the right-angled triangle PdF, Pd2 (y2)=FP-Fd2 (2a+2x2-2 a-x2)= 12 a x + 3x2; or y = 12 a x + 3 x2. This equation is that of the common hyperbola, whose axes have a given ratio commensurable in power, and therefore is very easily constructed in the following manner.

2

From a given point C, in a right line CD, draw two right lines CK, CL, indefinitely on each side, making with CD an angle of 60°, or with one another, an angle of 120°: Set off from C, the distance CV=2a; then between the two assymptotes CK, CL, and through the vertex V, construct an hyperbola, and this will be the curve required, or what is called by Mr. Winthrop, the trisecting curve.

*See Fig. 2, Plate 1.

For from V, erect the perpendiculars VG, VH; then the angles HCV, VCG, being each equal to 60°; HC, or CG, will be equal to 2 CV=4 a, and HV2=16 a24a2=12a; but by conics, CV (4 a2): HV (12a2):: 2 CV+Vdx Vd (4a+xxx): dP2; putting therefore dP, the ordinate of the curve, as usual, equal to y, we shall have y 12 a x + 3 x' for the equation showing the relation of the ordinates and abscissas of this curve, which is the same as that above determined for the trisecting curve; consequently, this is an hyperbola, whose assymptotes are inclined in an angle of 120 degrees, and whose axes are 4 a, 4 a 3, respectively.

October 2, 1809.

A STATEMENT

Of the Quantity of Rain which falls, on different Days of

the Moon.

BY JEREMIAH DAY,

PROFESSOR OF MATHEMATICS AND NATURAL PHILOSOPHY IN
YALE-COLLEGE.

THE

HE influence of the Moon upon the various bodies on the earth, is a subject of general observation. The swelling of the ocean has long been ascribed to this cause. There is also a very prevalent, though vague apprehension, that there is a connection between the vicissitudes of the moon, and the growth of vegetables, the progress of diseases, the changes of the atmosphere, and other important phenomena. It is desirable that so current an opinion should be brought to the test of accurate observation. Should it prove to be well founded, it might aid us, in predicting some of those changes in the atmosphere, and in the bodies around us, with which our daily concerns are intimately connected. To do justice to this subject, in all its extent, would require very numerous and diversified courses of experiments. For the purpose of ascertaining a single point, a calculation has been made on a series of observations, during the years 1804, 5, 6 and 7, on the quantity of rain which fell in New-Haven, in different periods of the moon.

The rain has been caught in a cylindrical vessel, ten inches in diameter, and about twenty inches deep. It is placed ten or twelve feet from the surface of the ground. The water, directly after it has fallen, is poured out, and measured in a tube one inch in diameter. In this way, as the area of the large cylinder is an hundred times as great as that of the small one, the depth of the water may be determined, to the thousandth part of an inch.

The snow is first melted, and then measured in the

same manner as the rain. This method is tolerably ac-
curate, except in some few cases of violent wind. In
these instances, the best expedient seems to be, to form
an estimate of the average depth of snow on the ground.

The quantity of water collected, whether from rain
or snow, is entered in a column against the day of the
month: and in an adjoining column, is noted the age of
the moon. The following table gives a view of the
whole quantities collected, on different days of the moon,
during forty-eight lunations; a period a little short of
four years.