119° 52' 15", and his alt, then = 37° 58' 51". At the uime of the second observation we find his azimuth from the north = 168° 20' 54", and alt. = 51° 20' 52". The azimuth from the north at the third observation = 73° 5' 19", and alt. = 6° 3' 56" ; from whence we find TA = 25.6165, TB = 15.9956, and TC = 188.2156 yards. Now in the triangle ATB, we have given AT, TB, and the 2 ATB,=71° 46' 51", to find the ¿ ABT = 71° 49' 42" ; again, in the triangle TBC we have TB, TC, and the 2 BTC = 95° 15' 35", to find the 2 TBC = 79° 56' 24" ; also the < ABT+ TBC = LABC = 151° 46' 6". Lastly, in the triangle ATC we have AT, TC and the 2 ATC= ATB + BTC, to find AC = 213.2017. Draw the diameter A D, and join CD; then by simp. geom. 16. 3. the angle CAD = ABC90° = 61° 46' 6" ; hence the diameter A D is found = 451 yards nearly, and the content of the field = 33 acres. Otherwise by Mr. J. Gawthrop, Leeds. . In this question we have given the time froni noon, the latitude and the sun's declin. to find his alt, and azimuths at each of those times : 1. Hour < 45° 0', declin. 11° 7 alt. 37° 58' 51" 50' 45", lat. 53° 5', to find.......... Sazim. 60° 7' 45'. 2. Hour 2 7° 30', declin. 14° 7 alt. 51° 20' 52" 53 22", lat. 53° 5', to find......... Sazim. 11° 39' 6". 3. Hour 2 100° 0', declin. 14° ( alt. 6° 3' 56' 58' 00", lat. 53° 5', to find .......... azim. 106° 54' 41", Now by adding the refraction and sun's semi-diameter = 15' 55", we have the apparent alt. of the sun's upper limb as follows: then the height 1. 38° 15' 54" 25.3562=length of the 2, 51° 37' 30" of the tree = 20 shadow at 9 A.M. 3. 6° 27' 36 x tan, of co. 15.8376% ditto 12 h. 176,6343= ditto 6h. 40m. Let T be the place of the tree; then in the triangle ATB we have AT 25.3562, BT = 15.8376, and the LT=71° 46' 51" to find the < TAB = 36° 24' 2'; and in the triangle BTC we have BT = 13.8376, TC =176.6343, and the < T = 95° 15' 35" to find BC = 178.6179 ; also in the triangle ATC, we AT alt. = 25.3562, TC = 176.6343, and the T = 167° 2' 26", to find TAC= 1° 20' 32", then TAB - TAC = BAC = 25° 3' 30". Draw the diam. AD, and join DC; then, DC A is a right 2, 31. Eu. 3, and the CDA = 25° 3' 30"; also AC= 178.6179, whence AD = 421.720 yards, and the area is found = 28a. 3r. 17p. In this solution the variation of the sun's declination, also refraction, &c. is taken notice of, which makes the result ditferent from the preceding one. 12. Qu. (42) Answered by Mr. J. Whitley. The figure being described as per question, divide A C in G, so that A G may have to AB the H P given ratio of P E to D B. Then G will be a given point, and GB G being drawn, meeting E D in F, A D will be given by position. B By sim. triangles, and the nature of the curve B PH, we have AG: AB :: DF: DB:: PE: DB; therefore P E = DF; consequently PD = FE; and the area BPH A generated by P D, will be equal to the area BGCB generated by FE = the quadrant A CB -A AGB. Again, the ordinate PD will be a maximum when its equal FE is so ; which is evidently the case when a tangent to the ellipse at the point E is parallel to GB; therefore, since G B is given in position, the point E may be found by Prob. 12. Book II. Simson's Conics, and from thence FE = PD, the greatest ordinate, will become known. Ergo Solutum. The same by Rylando. Put A B = t, AC = c,BD = x, DP = y, and DB : PE:: m : n. By the property of the ellipse, DE = v (2 x — x*), and, by the question, m:n::x: PE= ; therefore, DP = y = V (2tr x?) tnx which, when a maximum becomes ✓(2tr — x) nt? = o and by reduction : = it the ✓(2tr - x?)' greatest ordinate. nr nr m m cm că From the equation of the curve we have zy = nxx v (2tr for the fluxion of the area, the fluent of which, or general expression for the area of the curve, is * cir. seg. rad. t and versed sine - x2). m с nr? 2m ; To find the area geometrically ; since the ratio of DB: PE is given, the ratio of DB:D P is given also therefore, take DB : DP :: AB: AF, and draw BFG, then the area of the curve will be equal to the triangle A BG Again, by Messrs. Hine, Maffett, and Winward. Put AC = a, AB = b, m : n the given ratio, D B =x, and DP = y; then m :n :: *: EP= nr and, ጎ in b by the ellipse, a :b :: ✓ (2ax – 5): v (2 ar--*) a the correct fluent of which, when a = b, , is nb? where A is the quadrant of a circle whose rad. is a. 2m b being put = 0, by reduction we determine the greatest ordinate Also the fluxion of .v (22x — xo) — m as above. Mr. Gawthrop's solution to this question is very neat, and purely geometrical. Mr. Nesbits name was omitted among the answers to the 11 Qu. No. IV. and Mr. Putsey's to Qu. 10. We wish to observe to our correspondents in general, that as we have only a few days in which to collect and arrange all the materials for the Enquirer, it is next to impossible but that some trifling errors and omissions inust occur; we are as careful, and take as much pains as we can to avoid both, but we are human, and it will be sufficient, we hope, to observe that the errors and omissions are few, and the latter were never inten. tionally made. From the 20th to the end of the month is all the time we are allowed. Rylando is requested to send for any book he may think proper, the price of which does not exceed haif a guinea. NEW MATHEMATICAL QUESTIONS TO BE ANSWERED IN NO. VII. 1. Qu. (56) By Mr. Dunn, Broughton. Find three square numbers in arithmetical progression whose common difference is a cube number. 2. Qu. (57) By Mr. J. Baines, Horbury Bridge. What number is that whose square root, cube root, and biquadrate root are in arithmetical progression ? 3. Qu. (58) By Mr. Putsey, Pickering Academy. The radius of a circle inscribed in a right angled plane triangle is 2 feet ;--it is required to determine the sides of the triangle, such, that a heavy body may occupy one second of time in descending down the hy. pothenuse, when the plane of the triangle is perpendicular to the horizon ? 4. Qu. (59) By Mr. R. Maffett, Plymouth. Find two square numbers such that their sum, the root of the greater number added to the less, and the root of the less number added to the greater, may be all squares. 5 Qu. (60) 'By the sume. A person applied to a surveyor to measure him a field in the form of a right angled triangle; but the base and perpendicular being surrounded with water, he could only obtain the length of the hypothenuse = 100 chains, and that of a patii-way bisecting the angle at the base, and terminating in the perpendicular = 80 chains : now Mr. Cross-staff contends that from these data the area cannot be had, and he has bet the gentleman the price of a new field-book, that none of the Theodolite gentry who write to the Enquirer, can from thence find the content. Pray, gentlemen, show him that, for once, he is wrong. 6. Qu. (61) By Mr. J. Baines, Horbury Bridge. On what day of the year, in the lat. of 53° 45' N. will the shadow of the point of a gnomon describe an hyperbola, whose transverse diam. is to the conjugale as 9 to 11? 7. Qu. (62) By Rylando. To describe a parallelogram, which shall bave its area and perimeter respectively equal to the area and perimeter of a given triangle. 8. Qu. (63) By Mr. Pulsey. At the moment a heavy body was let fall perpendicularly from the top of an inclined plane, another body was projected directly up it from the bottom, with a velocity equal to that which the same body would have acquired by descending freely down the length of the plane ;--to determine their nearest approach; the length of the plane being 50, and its height 40 feet? 9. Qu. (64) By Pr. Jesse Winward, 2nd R. L. Militia. , What are of a circle is that, the product of whose sine into its versed-sine is the greatest possible ? 10. Qu. (65) By Mr. Roger Vayheeg. Porism. Having a circle given in magnitude and position, another circle may be found, given also in magnitude and position, so that if through a given point P, any line be drawn cutting the circles in A, D, and B, the solid under A P, P D, P B, may be always of the same given magnitude, |