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Let p +1 be the number of figures in the given number.
Subjoin to the right-hand of p,
2dly, Then the nearest whole number to one-third of the result, will be the power to which 2 has been raised, to produce the given number.
Ex. 1. What power of 2 is 2048?
Here the number of figure is 4; therefore
p14, and p = 3.
Since the given number begins with 2, we must subjoin 3 to the right side of p, when it becomes 33,
consequently 2048 is the 11th power of 2.
Ex. 2. What power of 2 is 4611686018427387904?
p+1= 19, and p = 18;
And since the given number begins with 4, we must subjoin 6 to p, when we have 186;
the given number is, therefore, the 62d power of 2.
The investigation of this rule is very easy:
Let the number given be N = A",
by logarithms we have
log. N= log. A” = n log. A ;
And since the number of figures in N is p + 1, log. N is nearly equal to p, and the first decimal of log. N will depend on the figure with which N begins. If it begins with 2, it will be 3; if it begins with 3, the decimal part will be 5
nearly, and so on, as in the rule. Let k be any one of these numbers, then
10 log. A
When A = 2, log. A = nearly, and the value of n is
10 p + k
as we have given in the rule.
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=3, and p = 2.
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
for the powers of 2; the reason of which is, that
3 is more 10
nearly equal to the logarithms of 2, than
is to that of 4.
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 +1 5, and p = 4;
and 15625 is the 6th power of 5.
This rule is easily proved by the formula already given, if
Any number, which is a power of 9, being given to ascertain 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 be the last figure but one of the given number: then,
If the given number ends with 1, it is the (10 t + 10 − ) power of 9, except = o, when it is simply the 10 t power of 9. If the given number ends with 9, it is the (10 t + i + 1)
Ex. 1. What power of 9 is 282429536481?
Here p + 1 = 12, and 10 is contained in 12 once; thereforet 1, also = 8: hence
10 t 10 - 1 = 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 to, also 16: then
and the given number is the 7th power of 9.
The investigation of this rule may easily be deduced, from considering the expansion of
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 situated at the distance of fourteen Portuguese leagues from the north of Lisbon, in the province of Estremadura, and comprehended in the Ouvidoria de Alenquer, 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.