nearly, and so on, as in the rule. Let k be any one of these numbers, then Given any number, which is a power of 4, to ascertain what power it is. Rule.-Proceed exactly as in the first part of the former rule; and, for the second part, Take the nearest whole number to one-sixth of the result, which will be the power to which the number 4 has been raised. Ex. 1. What power of 4 is 256 ? Here p + 1 = And since the given number begins with 2, we must subjoin 3; hence and 256 is, therefore, the 4th power of 4. Ex. 2. What power of 4 is 1152921504606846976 ? and the given number is the 30th power of 4. The investigation of this rule easily follows from that of the former; for And A, in this case, is equal to 4, consequently log. This rule is not quite so extensive as that which was given 3 10 for the powers of 2; the reason of which is, that is more 6 nearly equal to the logarithms of 2, than is to that of 4. 10 Given any number, which is a power of 5, to determine on inspection what power it is. Proceed exactly as in the two former rules, for the first part, and take the nearest whole number to one-seventh of the result; this is the power to which 5 has been raised. Ex. What power of 5 is 15625? Here p +15, and p = 4; and 15625 is the 6th power of 5. This rule is easily proved by the formula already given, if 7 we substitute in it which is nearly equal to the logarithm 10 Any number, which is a power of 9, being given to ascer⚫tain by inspection what power it is. All powers of 9 end either with 1 or with 9. p+1 being the number of figures in the given number, let t be the number of times 10 is contained in p + 1, and let I be the last figure but one of the given number: then, If the given number ends with 1, it is the (10+10) power of 9, except = o, when it is simply the 10t power of 9. If the given number ends with 9, it is the (10 t ++ 1) power. Ex. 1. What power of 9 is 282429536481 ? Here p+1 12, and 10 is contained in 12 once; there føre t = 1, also l≈ 8: hence = 10 t + 10 = 10 + 10— 8 = 12, and it is the 12th power of 9. Ex. 2. What power of 9 is 4782969 ? In this case we have p + 1 = 7, and 10 is not contained in 7; therefore t o, also l 6: then 10t+1+1=6+1=7, and the given number is the 7th power of 9. The investigation of this rule may easily be deduced, from considering the expansion of 9′′ — (10 — 1)". = If, however, the number, which is a power of 9, contains less than 22 places of figures, we may employ a shorter rule; for, in that case, the number of the figures it contains will be equal to the power to which 9 has been raised; thus 729 contain 3 figures: it is the 3d power of 9; so also 59049 consists of 5 figures, and it is the 5th power of 9. Any power of 11 being proposed, to discover on inspection what power it is, Take as many tens as there are contained in the number of figures, of which the given number consists, and add to them the last figure but one of the given number; the sum is the power to which 11 has been raised. Ex. What power of 11 is 14641 ? The number of figures is 5, in which 10 is not contained, and the last figure but one is 4: therefore 14641 is the 4th power of 11. These rules are, of course, only applicable to perfect powers of the respective numbers to which they relate, and there is considerable difference as to the extent to which they continue true; thus the 2d rule, given for the powers of 9, fails in the 23d power: whilst the rule for discovering the power of 11, will not, I believe, be found deficient for any power under the 240. ART. VII. Account of the Mineral Springs of Caldas de Rainha, in the North of Portugal, with an Analysis of the Water. By GEORGE RENNIE, Esq. London, Feb. 18, 1818. CALDAS is a small town, celebrated for its baths. It is situ ated at the distance of fourteen Portuguese leagues from the north of Lisbon, in the province of Estremadura, and comprehended in the Ouvidoria de Alemquer, which includes the villages of Alemquer, Alde Galega, da Merciar, Caldas Chamusca, Cintra, Obidos, and Selir de Porto, comprising a population of 30,000 souls. The surrounding country is well cultivated, and agreeably diversified by gentle inequalities of surface. The soil, which is sandy, reposes on red sand stone, covering a coal formation. The surrounding hills have no considerable elevation, they consist of coarse red sand and primitive limestones. The town, which is only remarkable for its baths, approaches the figure of an irregular square, and annually, though slowly augments. The houses are indifferently furnished, and the windows, for the most part, unglazed. Living is expensive, and the essential luxuries of life hardly obtainable. The accommodations at the inn, are, however, good. The hot springs are in the centre of the town, and inclosed by a neat substantial building, which is entered from the principal or western side, into a square vestibule. A room on the left constitutes a pharmacy. A dark vaulted passage conducts to the men's bath on the right, and a passage of about 30 feet long by 8 feet wide, connects the well-room with the vestibule. The interior of the hospital communicates with the well-room, where the water is administered by an attendant. I tried the temperature of the water when fresh drawn, it was then 85° Fahr. in the tumbler, when lowered into the well it indicated a variation of from 88° to 90° Fahr. The emission of vapour and sulphureous smell appeared to augment and diminish, but observed no regular intervals. The smell and taste was similar to that of Harrowgate water, but accompanied with less sparkling. Its volatile parts dissipated by exposure to the atmosphere, and formed a thin pellicle of sulphur on the surface. The colour of a bright silver table spoon was sensibly changed when suspended in the vapour, and the surface of a penknife was readily acted upon when immersed in the water. Boiling entirely deprived it of smell. The water is taken internally from 7 to 10 in the morning and evening, which is usually the time for bathing. The company employ these intervals in promenading to the dismal light of two or three lamps. The utmost precaution is observed in protecting the body by warm clothing, which is here very necessary. The insipidity of social intercourse, added to a total defect of public amusement, present no inducements to the residence of a stranger. The climate is subject to extreme variations of temperature, and the prevalence of violent winds in the spring, which blow uninterruptedly from the ocean. The excessive heat of the summer season in the day time, is succeeded by cold moist vapours in the evening, which render an exposure of the body extremely hazardous. On the 23d of August, the thermometer in the shade at mid-day, indicated 78° Fahr. and in the sun 115°, giving a difference of 37°. At midnight it was 44°, giving a difference of 34° between the temperature of mid-day and midnight. There are four baths appropriated to public use. But the principal one, or men's bath, is the one from which the sample was taken. It is 36 feet long, 9 feet wide, and about 2 feet 10 inches deep. It is enclosed by a spacious vaulted apartment. The spring oozes from the north-west corner, yielding upwards of 45 cubic feet per minute, calculating from the rise of water in the bath. A pump is placed in the source for the purpose of applying a stream on the different parts of the body. A thermometer plunged immediately into the source, indicated a temperature of from 93° to 94°, which agrees with the statement of Dr. Withering. The east end, which is most remote, varied by several degrees. A fine white sand over a stratum of argillaceous earth, covers the bottom |