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as most efficient. The public buildings consist of three national churches, an Episcopalian church, a court-house, and Tolbooth. The last is a handsome modern building with a fine tower terminated by a very elegant spire. The central school-house, situated upon the Green of Muirtown, is also a fine building, and comprises a large public hall, with six spacious apartments for the accommodation of the different classes and for the library and philosophical apparatus. Inverness is the centre of the custom-house district, which extends from the mouth of the Spey to Dornoch Frith on the east coast, and from Assynt Point to Ardnamurchan on the west. A striking alteration has of late taken place in the trade of grain; within fifteen years about 8000 to 10,000 bolls of oatmeal used to be imported annually into Inverness; while now from 4000 to 5000 bolls of oats are exported from its piers. . . . . The foreign annual imports into Inverness consist of from 400 to 600 tons of hemp, and three or four cargoes of timber or Archangel tar.' (New Statistical Account of Scotland.) There is no compulsory assessment for the support of the poor, who are provided for by special quarterly collections, by several charitable mortifications at the disposal of the magistrates, and from other sources. A short account of schools, which are numerous and upon the whole well conducted, is given in the article INVERNESS-SHIRE. The population of the burgh and parish of Inverness in 1831 was 14,324. The people in Inverness speak very good English: the tradition is that they learned it from Cromwell's soldiers. The climate of Inverness is much milder than might be supposed from its northern position in the island. Its mean annual temperature is about 47°, while that of the neighbourhood of London is about 48.5°, and that of London 50·5°. The mean annual quantity of rain which falls at Inverness is about 26:21 inches. This borough unites with Fortrose, Nairn, and Forres in returning one member to parliament.

chulish and in the bed of the river three miles south of
Fort William. Sandstone is also frequently met with.
The beds of the stratified rocks are usually highly inclined
to the horizon, approaching almost to the vertical, but the
dip varies. Their general direction is from south-west to
north-east. The two principal mountains are Ben Nevis
and Mealfourvounie. The former, which is separated from
the Grampians by the desolate tract called the Moor of
Rannoch, is composed of porphyry and granite, and rises
4374 feet above the level of the sea, being the highest
mountain in Great Britain. It is easily ascended on the
western side; and at about the height of 1500 feet the
prospect, till then confined, opens to the south-west and
discovers the Paps of Jura and several of the Hebrides.
Above the altitude of 2000 feet there is no vegetation, and
on the north-east side of the mountain, near its summit,
the snow lies throughout the year. Mealfourvounie, which
rises 2730 feet above the sea-level, is composed of a con-
glomerate rock and stratified sandstone, the latter of which
is of so hard a texture as to be used for the pavements
of the streets of Inverness. Some veins of lead and
silver have been discovered in several parts of the county,
and also iron ore in small quantities, but we are not aware
that mines have hitherto been worked to any extent. The
soil is for the most part light and sandy, with a subsoil of
gravel or clay; but in the neighbourhood of the town of
Inverness it is enriched by a fine loam deposited by the
waters of the adjoining frith.

Farms, Estates, and Agriculture.-In 1808 the landed property of this county was divided among 83 proprietors, viz.7 estates of the valued rents of 3000l. per annum (Scotch); 6 from 10007, to 30007.; 23 from 4007. to 10007.; 33 from 1007. to 4007.; and 14 under 1007. From that period to the present time we believe the above distribution has not undergone any material alteration. Formerly there were INVERNESS-SHIRE, a maritime county of Scotland, a great number of small arable farms only a few acres in bounded on the north by Ross-shire, on the south by the extent, but these have much decreased since the introducshires of Perth and Argyle, on the east by those of Nairn, tion of sheep farming. What remain of them are usuElgin, Banff, and Aberdeen, and on the west by the At- ally let from year to year, but the larger farms are lantic Ocean; the mainland is comprised between 56° 40′ let on lease, varying from seven to nineteen years. The and 57° 36′ N. lat., and between 3° 50′ and 5° 50′ W. long. farm-houses erected within the last forty years by the from Greenwich. Its greatest length from north-east to wealthier class of store farmers are for the most part well south-west is 88 miles, and its greatest width from north-constructed, but the dwellings of the cottagers and poorer west to south-east nearly 55 miles. According to Mr. tenants are described as being in every respect comfortless M'Culloch (Statistical Account of the British Empire the and mean. (McCulloch's British Empire, vol. i., p. 310.) entire county contains 4245 square miles, or 2,716,800 acres, The attention of the farmers is chiefly directed to the of which the mainland occupies 1,943,920, and its islands rearing of sheep and cattle. The sheep are mostly of the 773,760; the former having 84,480, and the latter 37,760 Cheviot and Linton breeds, and the stock at the present acres of water. It comprehends various districts, particu- time is estimated at 120,000; the stock of cattle is supposed larly that of Badenoch on the south-east, where it borders to amount to 40,000 or 45,000, and is chiefly of the Skye upon Perth and Aberdeenshire; Lochaber on the south ad- or Kyloe breed. In the month of July a fair for the sale of joining Argyleshire; Glenelg on the north-west bordering sheep is held annually at the town of Inverness, where, upon the ocean; besides many inferior districts, such as upon an average, 100,000 sheep and as many stones of Glengary, Glen Morrison, Glenshiel, &c. It also compre- wool are bought up for the southern markets. The labourers hends a considerable portion of the Hebrides, or Western and farm-servants generally live on potatoes with milk, and Isles, including the Isles of Skye, Harris, Benbecula, North oats and barley meal prepared in various ways, to which the and South Uist, Barra, &c. [HEBRIDES.] This county, wealthier tradesmen are able to add fish and butcher's which is extremely mountainous, is intersected by innumer- meat. The usual rate of ploughmen's and farm-servants' able lakes and rivers, and is divided into two nearly equal wages is 87. in money and six bolls of meal, with liberty to parts by the deep valley of Glenmore, which runs in a di- plant as much ground with potatoes as they can manure, rection from Fort William on the south-west to the town and female labour is commonly reckoned at two-thirds that of Inverness on the north-east. This county forms a large of men. The fields are frequently enclosed, and within the part of the Highlands of Scotland, and the general descrip- last twenty years a great deal of waste land has been tion of its geographical features cannot well be separated drained and reclaimed, and much ground planted; but from that of the division of the island to which it belongs. none of any consequence has been irrigated or embanked. [GREAT BRITAIN, p. 402.] By far the greater part of The average rent of cultivated ground varies from 17. to the surface is covered with heath, but a good deal of the 27. 10s. the acre, but in the immediate vicinity of the town heathy ground is arable, and a considerable extent of it has of Inverness it is as high as 57. to 77. the acre. been brought into cultivation during the present century. The population in 1831, according to the population returns for that year, was 94,797, of which 44,510 were males, and 50,287 females. The valued rent at the same period was 73,1887. Scotch, but the annual value of real property in 1815 was 185,5657. The county sends one member to parliament. [INVERNESS.]

Geology and Mineralogy. The prevailing rocks are of the primary class, having a highly crystalline structure, and being entirely destitute of organic remains. Gneiss is perhaps the most abundant, but huge masses of granite and of the oldest trap or porphyritic rocks are met with in the Grampians and the mountains of Glencoe and Ben Nevis. Limestone is found in several districts, and approaches to the nature of marble, particularly near Balla

Forests.-The fir woods in Glenmore and those of Strathspey in the adjoining county of Elgin are supposed to be more extensive than all the other natural woods in Scotland together. Glen Morrison, which opens into Glenmore, also contains much fine timber. In the parish of Kilmalie alone, near Fort William, it is estimated that there are about 14,000 acres covered with trees. Those which grow naturally are the oak, fir, birch, ash, mountain ash, holly, elm, hazel, and the Scotch poplar. Those which are planted are the larch, spruce, silver fir, beech, plane, and fruit trees. In these forests and the neighbouring mountains the herds of red and roe-deer roam in safety in recesses almost impenetrable to man. The alpine and common hare and other game are also abundant.

Manufactures.-Formerly a good deal of hemp, worsted,

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and linen yarn was made in this county, but this has greatly declined since the establishment of the large manufactories of the south. At the present time there is a hemp manufactory employing nearly 300 hands, and a woollen factory for the weaving of coarse clothing and Highland plaids and tartan. The produce of the former is principally exported to the London market and to the East and West Indies.

Inland Navigation and Roads.-The Glenmore, or 'Great glen of Albion,' as it is sometimes called, which stretches across the county from Fort William to the town of Inverness, is partially covered by three lakes, Loch Lochy, Loch Oich, and Loch Ness, which lie nearly in a straight line between the above-mentioned limits. Their aggregate length is 37 miles 704 yards, and the entire distance between Fort William and Inverness is 59 miles and 1628 yards. In 1802 Mr. Telford was appointed by the commissioners of the treasury to make a survey of these lakes and of the adjoining country preparatory to the cutting of a canal. His report was made in the following year, and the works were in full operation in 1805, but the whole line of navigation was not opened till the latter end of the year 1822. The expense of constructing the Caledonian Canal, as it is called, was defrayed by government. That part of the navigation which is not upon the lakes is 22 miles and 1628 yards in length: it is 50 feet wide at the bottom and 15 feet deep, though the original plan contemplated a depth of 20 feet. Loch Oich is the summit-level of the canal, and its elevation is 94 feet above the level of the sea on the east coast, at high water and ordinary spring tides. The entire cost was 986,924/., to which must be added a portion of the annual expenditure since the opening of the navigation, which has hitherto exceeded the produce of tonnage dues. The tonnage dues on vessels, whether laden or unladen, is one farthing per ton per mile, and produced in 1829 a revenue of 25751.; but the expenditure during the same year amounted to 45737., so that this canal promises to be but an unprofitable speculation. Its chief effect, as regards the town of Inverness, has hitherto been the commencement and gradual formation of a direct intercourse with the great western marts of Glasgow and Liverpool, and, through them, with the manufacturing districts with which these cities are so closely connected.' (New Statis. Acct.) The roads are under the management of the Parliamen1ary Commissioners for Highland Roads -a body appointed for opening the communication by land about the same time that the Caledonian Canal was projected. They are said to be maintained in a state of most efficient repair, the expense being defrayed partly by government and partly by contributions from the county proprietors. The principal rivers are the Spey, Ness, and Beauly; in all of which there valuable salmon fisheries, more particularly upon the Spey and Ness.

The parochial schools throughout the county are numerous and increasing, and the reader will find a very satisfactory description of their present state under the head of the several parishes in the above-cited work.

(New Statistical Account of Scotland; Playfair's Description of Scotland; Beauties of Scotland; Society's Map.of Scotland; Parliamentary Reports on the Caledonian Canal, 1803-4-5-6; Parliamentary Papers, &c.)

INVERSE, INVERSION. Any two operations of algebra are said to be inverse when one of them undoes, so to speak, the effect of the other; so that if both be successively performed upon the same quantity, the result is that quantity itself. For instance, the operations implied in 1+ and ✅ (x−1) are inverse to one another; for 1 + { √ (x − 1 }2 = x, √(1+x2 - 1) = x. We need do no more than rame addition and subtraction, multiplication and division, raising of powers and extraction of roots, as pairs of inverse operations.

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The operation of inversion is the solution of an equation, and vice versa. Let it be required to find the operation inverse to ox. Assume oxy, and find x in terms of y; say ay, then (y)=y, or and are inverse operations. Thus if 2 2 x = y, x = 1± √y+1, and either of the two, 1+ √ x + 1, or 1 +1, is inverse to x2 - 2x. It thus appears that a function may have more than one inverse function, and there are functions which have an infinite number; but there is a distinction by which one may be separated from all the rest. Let the Greek letters in this article be all functional symbols, or marks of operations to be performed, and let them come before the subject of operation, the quantity x, or y, &c., in the order in which they are to be performed. Thus apa denotes the result of performing the operation upon x, and then the operation a upon or. Now let or give x, where x is an unambiguous operation, and is, generally speaking, ambiguous, or presenting several different forms. Then and are inverse operations, and zz, and we might suppose at first that x=x; that is to say, we might imagine that destroys as well as that destroys 4. But since is ambiguous, it may be that only one or more of the forms of will satisfy xx, and not all: and that this will be the case with one is obvious, while we can show that it cannot happen with more than one. For though the same operation, performed on different functions, may produce the same function, yet different operations, performed on the same function, must produce different functions. If then a and ẞ be different forms of y, we have a x = x and


ßx = x; but we cannot have both a&x= x and B ф x are, where a and ẞ are different, or having absolutely the same form and value in both equations.

Education, Schools, &c.-Upon the formation, in the year 1818, of the Society for Educating the Poor in the Highlands,' a central or model school was erected on a large scale in the town of Inverness. This establishment, as might be expected, has already proved highly beneficial to the poor of the town and suburbs, and is upon an average attended by about 300 scholars. The same society supports

From all the inverses of a function px, then, we separate that one, ax, which gives both paxx and apx=x, and call it the convertible inverse. Its symbol is 1, so that means that operation which satisfies both the


inverse of a
the preceding



x = x and x=x. [EXPONENT.] In 2x: for 1+ √√(x2 − 2 x + 1) = 1 + x − 1 example 1+(x+1) is the convertible = x. But 1 √x2 - 2 x + 1 gives 1 − (x − 1) or 2 − x ; and we call this an inconvertible inverse.

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Every function which has more than one inverse is not only a function of x, but the same function of other functions of r. Let a be an inconvertible inverse of x; then a x is not x, let it be x. Then ax being x, a pris p x, or or is or, so that or is the same function of which it is of x. Thus in the preceding example x2 is the same function of 2 - x which it is of x; or


2 x

twelve other similar institutions of less extent in the more remote and thinly peopled part of the Highlands, and from its first establishment in 1818 to the 30th of September, 1834, its total expenditure amounted to 80237. The Raining School, founded in 1747 by Mr. John Raining of Norwich, and endowed by that gentleman with the sum of 10007, is superintended by two well qualified teachers, having salaries of 48.. and 40%. per year, together with a house and garden. The number of pupils is usually about 250. In addition to these, there are several private elementary schools, besides Sabbath evening-schools for religious inx2 - 2 x = (2 − x)2 — 2 (2 − x). struction, which are attended by a very considerable numWe have then this theorem: every function has as many ber of children. 'Inverness, thus already more amply different forms as inverses, and all these forms can be made provided than many other towns with the means of education by writing different functions of x instead of x in the original and improvement, has been further enriched by a munifi-function; and each inverse of the function is the converticent bequest of 10,000l. 3 per cent consols, left by the late Rev. Dr. Bell, the ingenious author of the Madras System of Education, and committed by him to the charge of the magistrates of Inverness, who contemplate, we understand, erecting another large charity-school, and relieving the Education Society of the burthen of supporting their central or model school on the Green of Muirtown.' (New Statistical Account.)

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The equation 4'x=x being understood, suppose that between the first and second operations we interpose the operation a, so that we have a 'x. This is no longer equal to x, but it is a function, the properties of which are closely connected with those of a x. For instance, if a x and ẞ be inverse to each other, then a p1xand ß1x are also inverse to each other: for a ßx=x and ‚aÞ ̃3þßþ ̄x, or $app-1x,

φαφ-(φβφ-1) απφαφ

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от ф ф x, or x. Thus knowing x + 1 and x 1 to be inverse functions, we know immediately that log (+1) and log (1) are inverse functions; and also(+1) and (x-1). For more detail on this subject see the article Calculus of Functions,' in the Encyclopædia Metropolitana.'

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INVERSION, in Music, is a change in the relative position of two sounds, or of the several notes of a chord. Thus c D, an interval of a 2nd, becomes by inversion (DC) a 7th. Example,

And CE G, the notes of the triad, or perfect chord, by inversion become the chord of the 6th (EG C), or of the 4th (GCE). Example,

For other musical Inversions, see CANON and FUGUE. INVOLUCRUM, in botany, is any collection of bracts round a cluster of flowers. In umbelliferous plants it consists of separate narrow bracts placed in a single whorl ; in many composite plants these organs are imbricated in several rows. If the bracts belong to a secondary series of the inflorescence, as in the partial umbels of an apiaceous plant, or in the solitary florets of Echinops, they form an involucel. The most singular state of the involucrum is that which is found in the genera Castanea, Fagus, Quercus, &c., where it forms a cup, or closed cover, remarkable in the European species of those genera, but much more so in the species of India.

INVOLUTE AND EVOLUTE (the curve unrolled and the curve from which it is unrolled), a name given to two curves so formed and placed, that supposing the second to be cut out from solid matter, the first can be formed by fastening one end of a thread upon a point in the second, attaching a pencil to the other end, and moving the pencil so that the thread may either gradually enwrap or be unwrapped from the curve to which it is fastened. Thus the pencil in the diagram is describing the involute of a circle, or the curve of which the circle is the evolute. But the evolute of a circle is evidently a point.

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If the line p P be drawn tangent to the evolute at p, it is one of the positions of the thread, and PT, the tangent of the involute at P, is perpendicular to pP. Also p P is the radius of curvature of the involute at P; this is to say, no circle can pass so near the curve at P, as the one which has p for its centre and p P for its radius. [CURVATURE.] Also, any arc of the evolute is the difference of two radii of curvature of the involute: thus the arc ap is the difference between a A and p P. Such are the principal geometrical connexions of the two curves.

Every curve has one evolute, and an infinite number of involutes. For instance, fastening the thread at b, and continuing it to M instead of A, we may with the cheeks ab and ba' produce another involute from them (represented by a dotted line); and any number, however great, by varying the position of M. But none of these involutes will be ellipses, except the one from which the evolute was made; though they will all be ovals having remarkable analogies with the ellipse. The proper name for curves described from the same evolute is parallel curves, since they have the fundamental property of parallel lines: for they never meet, though (if they admit of it) ever so far produced; a straight line perpendicular to one is always perpendicular to the other; and the part of the perpendicular intercepted is always of the same length. When arcs of parallel curves are required to be laid down, the most commodious method of proceeding is to construct the evolute of one of the arcs approximately, as follows. On the arc draw tangents at moderately small distances, and draw perpendiculars to those tangents. The parts of the tangents cut off from each by its neighbours will together give the arc of the evolute near enough for all purposes. And it may be well to notice that it will be a sufficiently accurate method of drawing the perpendicular to the tangent at a point P, if we take a small circle whose centre is P, bisect the arc A C B in C, and join and produce P C.

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The angular error thus committed is only a small portion

The following figure represents an ellipse with its evolute. of the angle made by the tangents at P and A.


Whenever the two arcs adjacent to a normal (or perpendicular to the tangent) of the involute are equal and similar, there is a cusp in the evolute; and the evolute generally recedes without limit as we approach a point of contrary flexure in the involute.

The mathematical method of finding the evolute is as follows. Let y=px be the equation of the involute, and let X and Y be the co-ordinates of the point on the evolute corresponding to that on the involute whose co-ordinates are x and y. Form the three equations


X-x+ (Y—y)=0;

1+ (



(dv)' – day (Y—y)=0;


parts, the first two of which have been long known, while the third, which contains the peculiar distinction of this method, is due to Mr. Horner.*

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1. In the article APPROXIMATION it is shown that if a be a value of x which makes or very small, then a (pa d'a) is a value of a which makes or much smaller; so that a continued succession of approximations may be made to a value of r which makes pr absolutely = 0. Here or means the differential coefficient or derived function and if x= Ax2 + Bæ”-1 + Сx2-2 + .. -2 ; then dr=nAr−+ (n −1)B−+ (n − 2)C-+...



2. Meaning by a root of ox, any value of which makes px=0, it is obvious that (x+a) is a function which has for its roots the roots of px, each diminished by a. And the substitution of x + a instead of x in the preceding value and from them eliminate x and y. The resulting equation of pe gives a well known development, of which an instance between X and Y is that of the evolute. But if the evolute will be more to our present purpose. Let the function be be given, and the involute is to be determined, let Y=ƒX AxBx+Cx2 + Dx2 + Ex + F .... (1). be the equation of the former, and from this and the latter Write x + a for x, and this becomes two of the preceding three equations eliminate X and Y. Ar + (5Aa + B) x* + (10Aa2 + 4Ba + C) x3 There will result a differential equation of the second order+(10Aa3+6Ba* +3Ca+D) x between y and x, the primitive of which is the equation of the involute, the two arbitrary constants being determined by the point at which the thread is supposed to be fixed and the length of the thread.

Thus if the curve be a parabola having the equation y=c, the equations for determining the evolute are

X-x+2cx (Y-y)=0;
1+4c2x2−2c (Y—y)=0;

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2c 2 2c

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the equation of the evolute of the parabola, which evolute therefore appears to be what is called a semi-cubical parabola.

For considerations similar to those which precede see CAUSTICS.

INVOLUTION and EVOLUTION. (Arithmetic.) Taking these words in their etymological sense, they might stand for the greater part of mathematical analysis. In their technical algebraical sense, they mean only the raising of powers, and the inverse operation, the extraction of roots. The revival however of a general process, accompanied by an improvement which makes it comparatively easy, renders it necessary to make a more extensive definition of the terms. We shall not relinquish any characteristic of the old meanings, and shall bring all corresponding processes together, by laying down the following definition:-Involution is the performance of any number of successive multiplications with the same multiplier, interrupted or not by additions or subtractions; and evolution is any method of finding out, from the result of an involution, what multiplier was employed, provided that the said method proceeds by involutions. Thus to determine 2x + 4x2 - 3x+10 by involution, we multiply 2 by x, and add 4; then multiply by and subtract 3, then multiply by r and add 10. If this give 1000, then any method of determining which proceeds by successive involutions is evolution.

A few years ago our only instances of evolution would have been common division, and the extraction of the square and cube roots, with references to Vieta, Harriot, Oughtred, and the older algebraists in general, for evolutionary methods of solving equations, bearing a strong likeness to such extractions. But since the publication of Mr. Horner's New Method of solving Equations of all orders,' Phil. Trans., 1819, the process which has rendered it worth while to propose the preceding extension of terms has been in the hands of mathematicians. For a more detailed account than we can here give, the reader is referred to the paper just cited, which is reprinted in the 'Ladies' Diary for 1838, or to The Theory and Solution of Algebraical Equations,' by Professor Young of Belfast (London, J. Souter, 1835).

We should begin with simple division, and the extraction of the square and cube roots, if we were writing an elementary treatise. But taking it for granted that the reader is familiar with the first two, at least, we shall proceed to describe the general process. This consists of three distinct

+(5Aa1+4Baa + 3Ca2+2Da + E) x + Aa3 + Ba1 + Ca3 + Da2+ Ea + F; which we may represent by

At+, a x + Þ ̧ a x3 + Þ2 a x2 + p' a x + $a. 3. The quantities da, p'a, pa, &c. may be determined by a succession of involutions, each one making use of the results of the preceding. Find pa by involution, of which the following are the steps:


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A If a be of only one significant figure (as 200, 6, '03), all the in the head, and we have thus (for the method is general, operations necessary to fill up this process can be performed though our example be only of the fifth degree) a working

Фа Фа φα φα φα

Mr. W. G. Horner was a schoolmaster and mathematical teacher residing at Bath, and died September 29, 1837. His works are announced as in preparation for the press, under the superintendence of Professor Davies, of the Roval Military Academy.

There has been some dispute about the right to the invention, of which we do not here speak in detail, as we have no doubt it will be extremely evident to all who examine the question that Mr. Horner is the first author and publisher (and, we believe, the only one) of that particular part of the method which goes beyond Vieta and lus successors. (See Companion to the Almanae' for 1859.)

method of answering the following question:-Given a certain equation or =0; required the equation = 0, the roots of which are each less by a than those of px = 0. If da came out =0, we should then know that a is a root of the equation: and the method of approximating to a root is as follows:-Suppose we have an equation of which the root (unknown to us) is 26.73. By trial, or otherwise, suppose we find that 20 is the highest denomination of the root, and we thereupon find another equation, each of whose roots is less by 20 than a root of the given equation: this is done by the preceding process, and one of the new roots (but unknown) is 6.73. If we can find that the highest denomination of this root is 6, we make another reduction of all the roots, and find a new equation, one of whose roots is 73. If we can then find 7 to be the highest denomination, we repeat the process and find an equation one of whose roots is 03. In finding the highest denomination of this root we find the root itself, evidenced by the pa of this final process being 0.


The first denomination of the root must be found by trial, or by some of the methods referred to in THEORY OF EQUATIONS. But the second and the remaining ones are found by comparing the results pa and p'a. If a be nearly a root,

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is still nearer. Consequently, by dividing pa by d'a, we may, after the second process, be sure of finding one figure of the remaining root correct. But after the first process we may be liable to an error of a unit (to be corrected by a new trial), as in extraction of the square root.

In order to obtain pa and not da, let the last coefficient, F, have its sign changed, and let the process in the column which contains it always be subtraction, and not addition. In the preceding type of calculation, we should then have -F


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Assuming 100 as a first approximation, we find that x+402x+60599x2+4059799x-5290748980 is an

equation having roots less by 100 than those of the given equation. And 529074898 contains 4059799 upwards of 130 times; but if any number of tens greater than 50 be taken, the accumulations of the next involution will give more than 5290, &c., as must be found by trial. Repeating the process, we find that +60223 +135899x+ 13634699-118087448=0 is an equation all whose roots are less by 50 than those of the last. We can now depend upon 118087448 divided by 13634699 giving one figure of

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The root of this equation is found to be 2:414213562, as follows. Beginning with the multiplier 2, one set of involutions brings us to the figures followed by colons, and x3+ 0x2-5x+2=0 is an equation on which the process is to be repeated. Dividing -2 by 5 we find that 4 is most probably the next figure, which is verified in the next trial, since the result of involution, 1936, is less than 2. We proceed in this way until 2:4142, containing half the number of figures wanted, is found, and this being a, we have found -0.000060831 for-pa, and -4485316 for p'a. The first divided by the second may be depended upon for doubling the number of figures, as commonly practised in the extraction of the square root. [APPROXIMATION.] The figures 13562 are found by a contracted division shown in the example.

But it is more convenient to avoid decimals in the process, which may be done as follows 1. If there be decimals in the coefficients of the equation, annex ciphers to every place in such manner that the number of decimals in the several places may be in increasing arithmetical progression. Then strike out the decimal points entirely, and proceed as with whole numbers, remembering that the root thus obtained will be 10 times too great if the progression increase by units, 100 times too great if it increase by twos, and so on. Thus 1812 ·6x2+33x+184 should be changed into 1·81x3-600x2+330000x+18'40000, and 181x-600x2+ 330000+1840000 will give ten times the required root. 2. When all the whole figures of the root have been obtained, and the decimal part is about to enter the calculation, before attempting to obtain the first decimal figure annex a cipher to the first working column on the left, two ciphers to the second, and so on to the end. Then proceed with the new figure as if it were a whole number, and make a new involution. When this is finished annex ciphers again as before. One additional advantage will be that the ciphers will serve to mark the places of completion of the individual involutions. If in any case pa should not contain d'a, place a cipher in the root, annex ciphers again, and then proceed. In some of the older algebraists, Oughtred for instance, the several vertical lines of figures are kept in their places by a set of ruled columns, the use of which is difficult. Mr. Horner has a similar contrivance; but the employment of ciphers removes all the difficulty, as in common division and the extraction of the square root. See the last example in this article. The method might easily be extended to the whole part of the root. The following is an instance of the method: ̧x1+x3 — x2 -2x-2=0.

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