ARITHMETICAL

, something relating to or after the manner of arithmetic.

Arithmetical Complement, of a logarithm, is what the logarithm wants of 10.00000 &c; and the easiest way to find it is, beginning at the left hand, to subtract every figure from 9, and the last from 10. So, the arithmetical complement of 8.2501396 is 1.7498604.—It is commonly used in trigonometrical calculations, when the first term of a proportion is not radius; in that case, adding all together, the logarithms of the 3d, 2d, and arithmetical complement of the 1st term.

Arithmetical Instruments, or Machines, are instruments for performing arithmetical computations; such as Napier's bones, seales, sliding rules, Pascal's machine, &c.

Arithmetical Mean, or Medium, is the middle term of three quantities in arithmetical progression; and is always equal to half the sum of the extremes. So, an arithmetical mean between 3 and 7, is 5; and between a and b, is (a+b)/2, or (1/2)a+(1/2)b.

Arithmetical Progression, is a series of three or more quantities that have all the same common difference: as 3, 5, 7, &c, which have the common difference 2; and a, a+d, a+2d, &c, which have all the same difference d.

In an arithmetical progression, the chief properties are these: 1st, The sum of any two terms, is equal to the sum of every other two that are taken at equal distances from the two former, and equal to double the middle term when there is one equally distant between those two: so, in the series 0, 1, 2, 3, 4, 5, 6, &c, twice 3 or 6.—2d, The sum of all the terms of any arithmetical progression, is equal to the sum of as many terms of which each is the arithmetical mean between the extremes; or equal to half the sum of the extremes multiplied by the number of terms: so, the sum of these ten terms 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, is (0+9)/2 X 10, or 9 X 5, which is 45: and the reason of this will appear by inverting the terms, setting them under the former terms, and adding each two together, which will make double the same series;

thus0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
inverted9, 8, 7, 6, 5, 4, 3, 2, 1, 0,
sums9, 9, 9, 9, 9, 9, 9, 9, 9 9;
where the double series being the same number of 9's, or sum of the extremes, the single series must be the half of that sum.—3d, The last, or any term, of such a series, is equal to the first term, with the product added of the common difference multiplied by 1 less than the number of terms, when the series ascends or increases; or the same product subtracted when the series descends or decreases: so, of the series 1, 2, 3, 4, &c, whose common difference is 1, the 50th term is 1+1X49, or 1+49, that is 50; and of the series 50, 49, 48, &c, the 50th term is 50-1X49, or 50-49, which is 1. Also, if a denote the least term,
z the greatest term,
d the common difference,
n the number of the terms,
and s the sum of them all;

then the principal properties are expressed by these equations, viz, Moreover, when the first term a, is 0 or nothing; the theorems become

and .

Arithmetical Proportion, is when the difference between two terms, is equal to the difference between other two terms. So, the four terms, 2, 4, 10, 12, are in arithmetical proportion, because the difference between 2 and 4, which is 2, is equal to the difference between 10 and 12.—The principal property, besides the above, and which indeed depends upon it, is this, that the sum of the first and last, is equal to the sum of the two means: so 2+12, or the sum of 2 and 12, is equal 4+10, or the sum of 4 and 10, which is 14. |

Arithmetical Ratio, is the same as the difference of any two terms: so, the arithmetical ratio of the series 2, 4, 6, 8, is 2; and the arithmetical ratio of a and b, is a-b.

Arithmetical Scales, a name given by M. de Buffon, in the Memoirs of the Acad. for 1741, to different progressions of numbers. according to which, arithmetical computations might be made. It has already been remarked above, that our common decuple scale of numbers was probably derived from the number of fingers on the two hands, by means of which the earliest and most natural mode of computation was performed; and that other scales of numbers, formed in a similar way, but of a different number of characters, have been devised; such as the binary and tetractic scales of arithmetic. In the memoir above cited, Buffon gives a short and simple method to find, at once, the manner of writing down a number given in any scale of numbers whatever; with remarks on different scales. The general effect of any number of characters, different from ten, is, that by a smaller number of oharacters, any given number would require more places of sigures to express or denote it by, but then arithmetical calculations, by multiplication and division, would be easier, as the small numbers 2, 3, 4, &c, are easier to use than the larger 7, 8, 9; and by employing more than ten characters, although any given number would be expressed by fewer of them, yet the calculations in arithmetic would be more difficult, as by the larger numbers 11, 12, 13, &c. It is therefore concluded, upon the whole, that the ordinary decuple scale is a good convenient medium amongst them all, the numbers expressed being tolerably short and compendious, and no single character representing too large a number. The same might also be said, and perhaps more, of a duodecimal scale, by twelve characters, which would express all numbers in a more compendious way than the decuple one, and yet no single character would represent a number too large to compute by; as is consirmed by the now common practice of extending the multiplication table, in school books, to 12 numbers or dimensions, each way, instead of 10; and every person is taught, with sufficient ease, to multiply and divide by 11 and 12 as easily as by 8 or 9 or 10. Another convenience might be added, namely, that the number 12 admitting of more submultiples than the number 10, there would be fewer expressions of interminate fractions in that way than in decimals. So that on all accounts, it is very probable that the duodecimal would be the best of any scale of numbers whatever.

Arithmetical Triangle. See Arithmetical TRIANGLE.

ARMED. A magnet or loadstone is said to be armed, when it is capped, cased, or set in iron or steel; to make it take up a greater weight; and also readily to distinguish its poles.

It is surprising, that a little iron fastened to the poles of a magnet, should so greatly improve its power, as to make it even 150 times stronger, or more, than it is naturally, or when unarmed. The effect however, it seems, is not uniform; but that some magnets, by arming, gain much more, and others much less, than one would expect; and that some magnets even lose some of their efficacy by arming. In general, however, the thickness of the iron armour ought to be nearly proportioned to the natural strength of the magnet; giving thick irons to a strong magnet, and to the weaker ones thinner: so that a magnet may easily be over-loaded.

The usual armour of a load-stone, in form of a rightangled parallelopipedon, consists of two thin pieces of iron or steel, of a square figure, and of a thickness proportioned to the goodness of the stone; the proper thickness being found by trials; always filing it thinner and thinner, till the effect be found to be the greatest possible.—The armour of a spherical load-stone, consists of two steel shells, fastened together by a joint, and covering a good part of the convexity of the stone. This also is to be filed away, till the effect is found to be the greatest.

Kircher, in his book de Magnete, says, that the best way to arm a load-stone, is to drill a hole through the stone, from pole to pole, in which is to be placed a steel rod of a moderate length: this rod, he asserts, will take up more weight at the end, than the stone itself when armed in the common way. And Gassendus and Cabæus prescribe the same method of arming. But Muschenbroek found, by repeated trials, that the usual armour already mentioned, is preferable to Kircher's; and he gives the following directions for preparing it. When, by means of steel filings and a small needle, the poles of a magnet have been discovered, he directs that the adjacent parts should be rubbed or ground into parallel planes, without shortening the polar axis; and the magnet may be afterwards shaped into the figure of a cube or parallelopipedon, or any other figure that may be more convenient. Plates of the softest iron are then prepared, of the same length and breadth with the whole polar sides of the magnet: the thickness of which plates, so as that they may admit and convey the greatest quantity of the magnetic virtue, is to be previously determined by experiment, in a manner which he prescribes for the purpose. A thicker piece of iron is to be annexed at right angles to these plates, which is called pes armaturæ, the foot or base of the armour: then the plates, nicely smoothed and polished, are to be firmly attached to each of the polar sides, whilst the thicker part or base is brought into close contact with the lower part of the magnet. In this way, he says, almost all the magnetic virtue issuing from the poles, enters into the armour, is directed to the base, and condensed by means of its roundness, so as to sustain the greatest weight of iron. Phys. Exper. and Geom. Dissert. 1729, pa. 131.

ARMILLARY Sphere, a name given to the artificial sphere, composed of a number of circles of metal, wood, or paper, which represent the several circles of the system of the world, put together in their natural order. It serves to assist the imagination to conceive the disposition of the heavens, and the motion of the celestial bodies.

This sphere is represented at Plate II, Fig. 6, where P and Q represent the poles of the world, AD the equator, EL the ecliptic and zodiac, PAGD the meridian, or the solstitial colure, T the earth, FG the tropic of cancer, HT the tropic of capricorn, MN the arctic circle, OV the antarctic, N and O the poles of the ecliptic, and RS the horizon.

The Armillary sphere constructed not long since by Dr. Long, in Pembroke-hall, Cambridge, is 18 feet in | diameter; and will contain more than 30 persons sitting within it, to view, as from a centre, the representation of the celestial spheres. The lower part of the sphere, which is not visible to England, is cut off; and the whole apparatus is so contrived, that it may be turned round with as little labour as is employed to wind up a common jack.

See also Mr. Ferguson's sphere in his Lectures, p. 194.

Armillary Trigonometer, an instrument first contrived by Mr. Mungo Murray, and improved by Mr. Ferguson, consisting of five semicircles; viz, meridian, vertical circle, horizon, hour circle, and equator; so adapted to each other by joints and hinges, and so divided and graduated, as to serve for expeditiously resolving many problems in astronomy, dialling, and spherical trigonometry. The drawing, description, and method of using it, may be seen in Ferguson's Tracts, pa. 80, &c.

ARTIFICIAL Numbers, Sines, Tangents, &c, are the same as the Logarithms of the natural numbers, sines, tangents, &c.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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ARISTARCHUS
ARISTOTELIAN
ARISTOTELIANS
ARISTOTLE
ARITHMETIC
* ARITHMETICAL
ARTILLERY
ASCENDANT
ASCENDING
ASCENSION
ASCENSIONAL Difference