twice the sine of the latitude of the place the secant of the Sun's altitude. 6. There are two places on the same meridian, whose latitudes are the complements of each other, and on a given day the Sun rises an hour sooner at one place than at the other. Required the latitudes of the two places. 7. If A and a be the altitudes of a star, on the same vertical circle on the same day, (d) the declination of the star, and (1) the latitude of 8. Construct a vertical south-east dial, for a given place. 9. When the Sun is in the equinoctial, the locus of the extremities of the shadow cast by a perpendicular object upon an horizontal plane is a straight line. 10. If a body revolve uniformly in a circular orbit, it is retained in that orbit by a force which tends to the centre of the circle; and if the periodic times in such circle a R", the force X 1 R 2 11. Determine the quantity of refraction by observations made upon the circumpolar stars. When will this method fail? 12. The parallax of a planet in right ascension being given, it is required to find the distance of the planet from the earth's centre, the earth being supposed spherical. 13. The sine of the excentric anomaly: the sine of the true anomaly: the radius vector: the semi-axis minor. 14. Find the distance of a planet from the Sun. 15. Suppose an eclipse of the Moon to last three hours; to how great a portion of the Earth will some part of it be visible? 16. Prove that when the first point of Aries, rises, the ecliptic makes the least angle of the horizon; and explain from thence the phenomenon of the harvest Moon. 17. When will the right ascension and declination of a star be diminished, and when increased, by the retrograde motion of the equinoctial points? 18. The equation of time arising from the obliquity of the ecliptic is a maximum, when the cosine of the Sun's declination is a mean proportional between the radius and the cosine of the obliquity. 19. Find the aberration of a star in latitude on a given day, and also the aberration of a planet in longitude. 20. Explain the method of determining the difference of longitude of two places on the Earth's surface, by means of a chronometer; and state the errors to which this method is liable. 8 + x; and extract the cube root of x6. of a 12-6r3. 3. Solve the following equations: VOL. I. T (a) mx+x=4b+2x (B) √4a+x=2√b+x— √x. 4. Insert (m) harmonic means between (a) and (b). 5. If, between all the terms of an arithmetical progression, the same number of arithmetic means be inserted, the new series will still form an arithmetical progression. 6. Sum the following series: 7, 5, 3, &c. to (5) terms 1+5+13+29+61 +&c. to (n) terms. of one sort (q) of 7. What is the number of permutations of (n) things, of which there are (p) another, and (r) of a third? total number of combinations admit of, =2"-1. And shew that the which (n) things 8. A, who travels only every other day, sets off from a certain place nine days after B, in order to overtake him, but travels four times as fast as B does. When will they come together? 9. Divide the number 35 into two such parts, that the square of the less divided by the difference of the two parts =45, 10. A person owes £150, to be paid at the end of nine months; and £60, at the end of six months. Required the equated time of payment, and investigate the rule. 11. ab is a ratio where (a) is prime to (b). They are the least in that proportion—and shew that a+ a very nearly in the. proportion of a+mx: a, when (x) is small compared with (a). m : |