mainder will be the common logarithm of the spherical excess in seconds and decimals. 5. Lastly, when the three sides of the triangle are given in feet; add to the logarithm of half their sum, the logs, of the three differences of those sides and that half sum, divide the total of these 4 logs, by -2, and from the quotient subtract the log 9-3267737; the remainder will be the logarithm of the spherical excess in seconds, &c. as before. One or other of these rules will apply to all cases ia which the spherical excess will be required.* PaOBLEM XII.. • , To determine the ratio of the earth's axes from the measures of convenient portions of a meridian in any two giyen latitudes; the earth being supposed a spheroid generated by the rotation of an ellipse upon its minor axis. The most accurate way of solving this problem, will be to compare, not merely single degrees measured on different parts of the meridian, but large portions of 5, 6, or 7 degrees, the most extensive that have been correctly measured; according to the method proposed by Professor Play fair (Edinburgh Transactions, vol. v.), which is as below. Let the ellipse pEptt represent a terrestrial meridian passing through the poles r,/>, and cutting the equator in E, o. Let c be the centre of the earth, Cq the radius of the equator = a, and Pc, half the polar axis = b. .' ""1 • The intelligent student who wishes to go more minutely into the subject of geodesic operations, especially in reference to thfc determination of the figure and magnitude of the eiirlh, may consult the chapter on that subject in the 3d vol. of Dr. Mutton-* Course, Colonel Madge-s " Account of the Trigonometrical Survey of Kngland and Wale?," M. Puissant-s works, entitled "Geodesic" and "Trailed de Topographic, d'Arpentage, &c." Mechain and Delambre, " Base du Systiihe Melrique Decimal," and chap. xizv. of Uelambre-f quarto " Aitroaomie." Let AB.be any small arc of the meridian, having its centre -of curvature in H; join Ha, Hb, intersecting Cq in K and L. Let <p be the naeasure of the latitude of A, or the measure of the angle Qka, expressed in decimals of the radius 1; not in degrees and minutes. Then, the excess of the angle Qlb above Qka, that is, the angle Lhk or Bha will be = $<p. Also, if the elliptic arc Qa = z, then will Ab = £z — ^ x Ah. . Now it is shown by the writers on conic sections, that the radius of curvature at, A, that is, Note. From these theorems, Professor Playfair, by comparing an arch of 3° 7' 1" measured in Peru, with an 7 arc of 8° 20' 2^" measured from Dunkirk to Perpignan, found - = -57:^ nearly. And from the same theorems, a JOO I, by comparing the arc in Peru with the arc from Dunnose to Clif-Von, in Yorkshire, amounting to 2° 50' 23}", obtained 8 for the resulting compression. ( See my Collection of Dissertations and Letters relating to the Trigonometrical Survey of England and Wales, pa. 4-7.) There is great reason to conclude that the true compression lies between these limits. Section II. 1 . Demonstrate the truth of the following analogy, viz. As the sine of half the difference of two arcs which together make 60° or 90° respectively, is to the difference of their sines; Bo is 1 to */ 3 or */ 2, respectively. 2". Demonstrate that 4 times the rectangle of the sines of two arcs, is equal to the difference of the squares •f the chords of the sum and difference of those arcs. 3. Demonstrate that of any arc A, _ 1 - tan A Sin A ~~ V (I + cot2 A) y' (1 + tan" A)" 4. Also, that' 1 Cot A COSA= • 16 — iiO sec' A + 5 sec*A .- ,• 10t Demonstrate that arc to tan £ 4- arc to tan f = arc 45° arc to tan y + 2 arc to tan f = arc 45°. "Hi Demonstrate that the tangent of the lum. of any number of arcs will be represented by '; A — C + K — G ftc. the sum of all the tangents of the separate arcs being de« noted by A, the sum of all their rectangles by B, the sum of all their solids by c, &c. 12. Find the arc whose secant and co-tangent shall be equal. Ans. Arc whose sine is J J 5 — \, or 38° 10'. 13. Find the arc whose sine added to its cosine shall be equal to a; and show the limits of possibility. .." Ans. a2 must never exceed 2. 14°. What arc is that wliose tangent and cotangent shall together be equal to four times radius? Ans. Arc of 75* or of 15°. |