EXTERMINATION
, or EXTERMINATING, in Algebra, is the taking away, or expelling of something from an expression, or from an equation: as to Exterminate surds, fractions, or any particular letter or quantity out of equations.
Thus, to take away the fractional form from this equation ; multiply each numerator by the other's denominator, and the equation becomes a2d + dx2 = 2abc, out of fractions.
Also, to take away the radicality from the equation , raise each to the 2d power, and it becomes 9a2-9x2 = 4c2.
For Exterminating any quantity out of equations, there are various rules and methods, according to the form of the equations; of which many excellent specimens may be seen in Newton's Algebra, pa. 60, ed. 1738; or in Maclaurin's Algebra, part 1, chap. 12. For example, to Exterminate y out of these two equations, ; subtract the upper equation from the under, so shall there arise 3b-a-x=2x-b; then, by the known methods of transposition &c, there is obtained 4b-a=3x, and hence x = (4b-a)/3.
EXTERNAL Angles, are the angles formed withoutside of a figure, by producing its sides out.
In a triangle, any External angle is equal to the sum of both the two internal opposite angles taken together: and, in any right-lined figure, the sum of all the external angles, is always equal to 4 right angles.
EXTRA-Constellary Stars, such as are not properly included in any constellation.
EXTRA-Mundane Space, is the infinite, empty, void space, which is by some supposed to be extended beyond the bounds of the universe, and consequently in which there is really nothing at all.
EXTRACTION of Roots, is the finding the roots of given numbers, or quantities, or equations.
The roots of quantities are denominated from their powers; as the square or 2d root, the cubic or 3d root, the biquadratic or 4th root, the 5th root, &c; which are the roots of the 2d, 3d, 4th, 5th, &c powers. The Extraction of roots has always made a part of arithmetical calculation, at least as far back as the composition of powers has been known: for the composition of powers always led to their resolution, or Extraction of roots, which is performed by the rules exactly reverse of the former. Thus, if any root be con- | sidered as consisting of two parts a + x, of which the former a is known, and the latter x unknown, then the square of this root being a2 + 2ax + x2, which is its composition, this indicated the method of resolution, so as to find out the unknown part x; for having subtracted the nearest square a2 from the given quantity, there remains 2ax + x2 or ―(2a + x) X x; therefore divide this remainder by 2a, the double of the first member of the root, the quotient will be nearly x the other member; then to 2a add this quotient x, and multiply the sum 2a + x by x, and the product will make up the remaining part 2ax+x2 of the given power.
The composition of the cubic or 3d power next presented itself, which consists of these four terms a3 + 3a2x + 3ax2 + x3; by means of which the cubic roots of numbers have been extracted; viz, by subtracting the nearest cube a3 from the given power, dividing the remainder by 3a2, which gives x nearly for the quotient; then completing the divisor up to 3a2 + 3ax+x2, multiply it by x for the other part of the power to be subtracted. And this was the extent of the <*>xtraction of roots in the time of Lucas de Burgo, who, from 1470 to 1500, wrote several pieces on arithmetic and algebra, which were the first works of this kind printed in Europe.
It was not long however before the nature and composition of all the higher powers became known, and general tables of coefficients formed for raising them; the first of which is contained in Stifelius's arithmetic, printed at Norimberg in 1543, where he fully explains their use in Extracting the roots of all powers whatever, by methods similar to those for the square and cubic roots, as above described; and thus completed the Extraction of all sorts of roots of numbers, at least so far as respects that method of resolution. Since that time, however, many new methods of Extraction have been devised, as well as improvements made in the old way.
The Extraction of roots of equations followed closely that of known numbers. In De Burgo's time they extracted the roots of quadratic equations, the same way as at present. Ferreus, Tartalea, and Cardan extracted the roots of cubic equations, by general rules. Soon afterwards the roots of higher equations were extracted, at least in numbers, by approximation. And the late improvements in analytics have furnished general rules for Extracting the roots, in infinite series, of all equations whatever. All which methods may be seen in most books of arithmetic and algebra. Of which it may suffice to give here a short specimen of some of the easiest rules for Extracting the roots of quantities and equations, as they here follow.
I. To Extract the Square Root of any Number.— Point off, or divide the number, from the place of units, into portions of two figures each, as here of the number 99856, setting a point or mark over the space between each portion of two sigures. Then, beginning at the left hand, take the greatest root 3, of the first part 9, placing it on the xight hand for the first figure of the root, and subtracting its square 9 from the said first part; to the re- mainder, which here is o, bring down the 2d part 98, and on the left hand of it place 6 the double of the first figure 3, for a divisor; conceive a cipher added to this, making it 60, and then divide the 98 by the 60, the quotient is 1 for the second figure of the root, which is accordingly placed there, after the 3, also in the divisor after the 6 and below the same; then multiply these as they stand, the 61 by the 1, and the product 61 set below the 98, and subtract it from the same, which leaves 37 for the next remainder; to this bring down the 3d period 56, making 3756 for the next resolvend: then form its divisor as before, viz, doubling the root 31, or adding, as they stand in the divisor, the 1 to the 61, either way making 62, which with a cipher makes 620, by which divide the resolvend 3756; the quotient of this division is 6, to be placed, as before, both as the next figure of the root, and at the end of the divisor 62, and below itself there; then multiply as they stand the whole divisor 626 by the 6, the product 3756 is exactly the same as the resolvend, and therefore the number 316 is accurately the square root of the given number 99856, as required.
When the root is to be carried into deccimals, couplets of ci- phers are to be added, instead of figures, as far as may be wanted. In which case too, a good abbreviation is made, after the work has been carried on to half the number of figures, by continuing it to the other half only by the contracted way of division; as here in the annexed example for the square root of 2 to eight decimals, or nine places of figures in all.
II. To extract the cubic root, or any other root whatever. This is easiest done by one general rule, which I have invented, and published in my Tracts, vol. 1, pa. 49, which is to this essect: Let N be any number or power, whose nth root is to be extracted; and let R be the nearest rational root of N, of the same kind, or Rn the nearest rational power to N, either greater or less than it; then shall the true root be very nearly equal to ; which rule is general for any root whose index is denoted by n. And by expounding n successively by all the numbers 2, 3, 4, 5, &c, this theorem will give the following particular rules for the several roots, viz, the
| 2d or squ. root, | (3N+R2)/(N+3R2) | X R; | ||
| 3d or cube root, | (4N+2R3)/(2N+4R3) | X R, or | (2N+ R3)/(N+2R3) | X R; |
| 4th root | (5N+3R4)/(3N+5R4) | X R; | ||
| 5th root | (6N+4R5)/(4N+6R5) | X R, or | (3N+2R<*>)/(2N+3R<*>) | X R; |
| 6th root | (7N+5R6)/(5N+7R6) | X R; | ||
| 7th root | (8N+6R7)/(6N+8R7) | X R, or | (4N+3R7)/(3N+4R7) | X R; |
| &c. | &c. |
For ex. suppose the problem proposed, of doubling the cube, or to sind the cube root of the number 2. Here N = 2, n = 3, and the nearest power, and root too, is 1: Hence ; then the first approximation.
Again, taking R = 5/4, and conseq. R3 = 125/64 : Hence ; then , for the cube root of 2, which is exact in the very last figure.
And again by taking 635/504 for the value of R, a great many more figures may be found.
III. To Extract the Roots of Algebraic Quantities.— This is done by the same rules, and in the same manner as for the roots of numbers in arithmetic, as above taught. Thus, to Extract the square root of 4a2 + 12ax + 9x2.
| 4a2+ | 12ax+9x2 (2a+3x the root | |
| 4a2 | ||
| 4a+3x | 12ax+9x2 | |
| 3x | 12ax+9x2 |
So also the root is carried out in an infinite series, in imitation of the like Extraction of numbers in infinite decimals: thus, for the square root of a2+x2.
To extract the cube root of a3-x3 by the general rule in the 2d article.—Here ; therefore, by the rule, &c, which is the cube root of a3-x3 very nearly.
But these sorts of roots are best extracted by the Binomial Theorem; which see.
IV. To Extract the Roots of Equations.—This is the same thing as to find the value of the unknown quantity in an equation; which is effected by various means, depending on the form of the equation, and the height of the highest power of the unknown quantity in it: for which, see the respective terms, Equation, Root, Quadratic, Cubic, &c.
The most general, as well as the most easy, method of Extracting the roots of all equations, is by Double Position, or Trial-and-Error; as it easily applies to all sorts of equations whatever, be they ever so complex, even logarithmic and exponential ones. There are also several other good methods of approximating to the roots of equations, given by Newton, Halley, Raphson, De Moivre, &c; of which the most general is a rule for Extracting the root of the following indefinite equation,
viz, , given by M. De Moivre in the Philos. Trans. vol. 20. p. 190, or Abr. vol. 1, pa. 101.
