INULIN, a peculiar vegetable substance which is spon taneously deposited from a decoction of the roots of the Inula Helenium. It is a white powder, like starch, is insoluble in cold and soluble in hot water, from which it is deposited on cooling, and this distinguishes it from starch. With iodine it gives a greenish-yellow compound, which is not permanent. Inulin is distinguished from gum by its insolubility in cold water, and by not giving saccholactic acid when digested in nitric acid.

imagine, and therefore cannot require a greater. For a man cannot conceive of a greater certainty than that any idea in his mind is such as he perceives it to be, and that two ideas, wherein he perceives a difference, are different and not precisely the same.' His definition, or rather explanation, of intuition is as follows:-'Sometimes the mind perceives the agreement or disagreement of two ideas immediately by themselves, and this, I think, we may call intuitive knowledge. In this the mind is at no pains of proving or examining, but perceives the truth as the eye does the INVARIABLE (Mathematics), the same word in meanlight, only by being directed to it.' (Essay on Human Un- ing as CONSTANT, which see. There are however two sorts derstanding, b. iv., c. ii., §1.) Campbell's definition is of constants, which it is desirable to treat under different similar having defined truth to be the conformity of our names: the first, which we may call a constant, or a comconceptions to their antetypes in the nature of things, he mon constant, meaning a quantity which is absolutely indeclares intuitive truth to be that which is perceived imme-variable; the second meaning a function which may vary, diately on a bare attention to the ideas under review.' but which does not vary in the processes required by a given equation. This we propose to call the invariable function of that equation, or its invariable.

The nature of the relation which subsists between intuition and reasoning has been strongly contested. While Beattie maintains that the connexion between them, how Thus, in a common differential equation, which is supclosely soever they are found in general to be connected, is posed to be true of y and x when a passes through all not necessary, but, on the contrary, a being endued with stages of magnitude whatsoever, the only invariable is an one may be destitute of the other; Dugald Stewart, on absolute invariable, or a common constant. But in an the other hand, insists that the two are not radically distinct, equation of differences, in which only passes from one although by most writers they are considered to be different whole number to another, the invariable function is any faculties. Locke having rightly maintained that every one which remains unaltered by changing a from one step which the reason makes in demonstrative knowledge whole number to another. Thus, [INTEGRATION, FINITE] has intuitive certainty, and that consequently the power of instead of saying that the solution of Ay = x + 1 is reasoning presupposes that of intuition, Stewart thinks(x+x) + C, where C is a constant, we may allow C to that the intuition of Locke implies the power of reasoning; be any function of x, which is unaltered by changing x or, at least, that intuition combined with memory explains from one whole number to another. Such a function is reasoning. Here his usual sagacity appears to have failed (cos. 2x), so that the solution is (x2+x) + (cos. 2πx), Stewart. While the mind itself is perfectly simple, it and the last term is the invariable of the equation. has been, for the purpose of attaining accuracy of language Again, suppose it required to solve the functional equaand distinctness of theory, supposed to be multiple; and tion p (x2)=2 px. One solution of this is x = clog x, distinct faculties have been ascribed to it according as its where c is any absolute constant. But the equation is several operations comprise more or fewer elements. Ac solved if c be a function of x, provided it be one which does cording therefore to his own account, reason, which involves not change when x is changed into x. Such a function is the element of time, must be kept distinct from intuition, which does not involve that element. {2π or any function of it,

or pa

Cos.

log. log

log 2

any function of cos.

{2π

log, loglog x. log 2

The proper objects of intuitive certainty are identical propositions. This of course does not mean propositions verbally identical; such as 'a man is a man. But while the object of thought is perfectly and always one, it may present itself to the thought under a variety of aspects, either General methods of finding invariable functions, as far as dissolved into its elements or as combined into a whole. It they have yet been given, will be found in the Encyclopædia is this identity under an apparent diversity that constitutes Metropolitana,' article 'Calculus of Functions.' that original and primary evidence which makes certain INVENTION. This term, when used in the language propositions, as soon as the respective terms are understood, of art, has a different signification from what it usually bears to be perceived intuitively. On the other hand, the appa- in common language. It does not mean discovery, but rent identity of a real diversity is the ground of all sophis- combines conception, or the peculiar way in which an artist's tical argument. The ultimate form of legitimate argumen-mind takes cognizance of a subject to be represented, with tation is, a = b, b = c, ... a = c. But every fallacy, when the mode of treatment, or choice of objects and manner of detected, will invariably be found to be a = b±r, b = =c, disposing them best adapted for producing a desired effect. ... a = c. The sophistry consists in the suppression of the Thus, in painting and sculpture, it is the faculty by which element r, either positive or negative. the most perfect mode of illustration, by colour or by form, is suggested to the artist, and by which the mind of the spectator is led to comprehend the truth, the intention, and the whole purpose of the work before him; but so distinct is it at the same time from perfect execution, that it is often found to exist independently of excellence in that particular, some of the finest inventions in art being manifestly defective in technical requirements. It is therefore the highest quality in the constitution of the artist's mind; as Opie says, 'Destitute of invention, a poet is but a plagiary, and a painter a copier of others.' (Lectures on Painting.)

In the philosophy of Kant the term intuition (anschauung) is used to denote the single act of the sense upon outward objects according to its own laws. It appears to be employed in a like sense in the following extract from Glanvil-Some say that the soul is not passive under the material phantasms; but doth only intuitively view them by the necessity of its own nature, and so observes other things in these their representatives.' (Vanity of Dogmatising, c. iv., p. 29.)

It is hardly necessary to enter into the question whether the power of invention be a primary and original law of the mind, or whether the effect of cultivation. Some have be

I'NULA, a genus of composite plants, one of whose species, I. Helenium, is used medicinally. This plant is a native of various parts of Europe, in pastures and woods; it has a thick bitter mucilaginous root, a stout stem three feet high, broad ovate serrated leaves, and large yellow flower-lieved it may be a result of acquirements begun in youth, heads, which are solitary at the end of the ramifications.

and carried on till the power is developed and perfected; others conceive that it is unatatinable by any human effort, and is part of the original constitution of the mind.

But even admitting invention to be a gift of nature, and not

I'NULA HELENIUM (Elecampane), an indigenous perennial herbaceous plant, found in moist meadows, the root of which is used in medicine. This part is thick and branching, brown externally, white internally, with an aro-reducible to rule, nor to be taught by any regular process, matic odour and a mucilaginous taste, at first bitter, afterwards sharp and camphor-like. In addition to mucilage and a large quantity of a variety of starch termed inulin, it contains a crystallized volatile oil (stearopten), a bitter extractive, an acrid resin, and some salts of lime, &c.

it still may be improved by study. Whatever natural disposition or original capacity may exist-and it will not, we suppose, be denied that some minds are more bountifully endowed than others-every power short of creation must have groundwork and foundation on which and out of These ingredients give it a tonic and stimulating pro- which to exercise itself; and even the inventive faculty, perty, and it is employed in debility of the stomach, and which seems to approach nearest to creation, depends upon other diseases of mucous surfaces unattended with inflam-knowledge, by whatever means acquired, for materials with mation. It is however not much used.

which to develop and declare itself. Sir Joshua Reynolds

(Discourses) says, 'He who has the most materials has the greatest means of invention, and if he has not the power of using them, it must be from a feebleness of intellect; and it is in vain to endeavour to invent without materials on which the mind may work,' &c.

tions of artists, and discriminate between the efforts of elevated and original minds and the commonplace performances of mere mechanical copiers. Invention is required in every branch of art to raise it above tameness and insipidity: it is indeed the magic power by which works of art first attract and then fix the attention.

Raffaelle, by the wonderful ability and power which he has shown in choosing subjects in which the greatest quan- It is hardly necessary to observe, that, difficult as it may tity of matter or incident could be introduced, and then in be to prescribe bounds to the imagination or the power of representing them at the most critical moment for illustration, invention, it has in art certain and defined limits beyond in combining all the most striking and affecting circum-which the painter and sculptor should not attempt to venstances, and filling the spectator's mind with the whole story, ture. When the artist dashes into extravagance, defies or by bringing before him, as it were, the past, the present, and outrages nature, and, with a view of exciting wonder, steps even suggesting that which is to follow, may justly be consi- out of the region of what is, has been, or may be, he only dered the greatest master in invention. He was gifted, if any shows that he has been gifted with fancy, but that it is man ever was, with the fullest portion of natural and inhe-wild and ill-regulated; he may awaken surprise, and may rent genius, but he attained his eminence by the most perse- mistake it for admiration, but he will produce no lasting vering course of exercise and observation, as the necessary nor beneficial impression, and his undisciplined fantasy will and only means through which the inventive faculty could never deserve to be ranked with the genius that has nobly be manifested. He studied nature diligently and profoundly illustrated nature by the only just, safe, and legitimate in all her varieties of beauty and expression. Nothing means, namely, her own beautiful, and expressive, and seems to have escaped him; everything that offered itself perfect works. out of her great storehouse was treasured as serviceable to INVENTORY. [EXECUTOR.] his art, and he acquired such an accumulation of materials, INVERARY, a royal burgh and seaport, capital of the serving as handmaids to his invention, that whatever sub-county of Argyle, situated on a small bay at the head of ject came before him found him prepared, and was imme- Loch Fyne, where the river Aray falls into that arm of the diately dignified with all the expression, truth, propriety, sea, 75 miles west by north from Edinburgh. The town and completeness, if we may use the word, that it was was erected into a royal_burgh by charter granted by gapable of receiving. Raffaelle never reached the perfect Charles I. and dated 28th January, 1648. (Municipal Corbeauty and character almost superhuman which appear in poration Reports.) The whole territory, with the exception the finest works of the Greeks, nor, in colour, the magic of a small feu, is the property of the Duke of Argyle, of brilliancy and breadth of Titian, another master-spirit; yet, whom the inhabitants hold their houses and grounds either in the largest and most comprehensive sense of the quality under leases or as tenants at will. It is governed by two we have been describing, he stands (perhaps with one baillies and nine common-councillors. The annual income mighty exception) without an equal or a rival. of the burgh is about 1807. and the annual expenditure is The examples which may be most satisfactorily adduced somewhat less. The town consists chiefly of one row of in illustration of invention in the fine arts, both for their houses facing the bay, built with great uniformity and excellence and for the facility of reference, as we are so covered with slate. The arrangements for watching, cleanfortunate as to possess them in this country, are the Car-ing, lighting, and the supplying of water are confided to the toons of Raffaelle preserved at Hampton Court. Of these the Paul preaching at Athens,' The Sacrifice at Lystra, and 'The Death of Ananias,' may be selected as the most remarkable for the quality we have been considering. Equally admirable, though totally in a different style, the frescoes of Michael Angelo, in the Sistine Chapel at Rome, must be quoted as triumphs of invention, a proud achieve. ment of the human mind. The comprehensiveness of his scheme of illustration, with the greatness and energetic character of his design and composition, render this one of the finest monuments that art has to boast. In viewing the magnificent works of these two masters, namely, of M. A. Buonarotti, in this chapel, and of Raffaelle d'Urbino, in the loggie and stanze of the same palace (the Vatican), the spectator has a series of examples of as wonderful efforts of inventive genius in historical design as it seems possible to produce. The works of Rubens offer also fine examples of invention, though the quality of his design, or rather of his forms, was not according to a classical or pure standard. It should be observed here that invention is quite independent of the class of design; its force and power INVERNESS, a seaport town and royal burgh of some may be displayed in every part of the art, and in subjects antiquity, the capital of the county of Inverness, and the prinof inferior grade, or even in the mode of treating colour, cipal town of the Highlands. It is situated at the southern light, and shade. Rembrandt, to proceed with further extremity of the Moray Frith near the eastern entrance of illustration, is one of those who displayed very high powers the Caledonian Canal, 155 miles north by west from Edinof invention; a genius,' Fuseli says, of the first class in burgh. The earliest charters upon record are those of King whatever relates not to form;' and he justly eulogises his William the Lion, four in number, conferring several privi'powers of nature' and 'the grandeur, pathos, and simpli- leges upon the burgesses, which were confirmed and extended city of his composition.' Thus also, though the quality of by the subsequent charters of Alexander II., III., Robert I., his art was not of the highest or grand class, the merit David II., James II., Queen Mary, and James VI. The last of invention is eminently due to our own Hogarth. constitutes the governing charter of the town, and is dated 1st Opie, in speaking of this artist, alludes in terms of high January, 1591. (Municipal Corporation Reports.) The admiration to a fine example of invention in one of his pic-management of the affairs of the burgh is vested in a provost, tures of the series called The Rake's Progress.' In the bagnio scene he has introduced in the back-ground one of the dissolute women of the party setting fire to a map of the World.

We have referred only to a very few out of the numerous artists whose works are worthy of attention as examples of invention; and have confined ourselves to some of the leading painters, though we might easily multiply them from productions in the sister art. Enough however has been said to point out the nature and value of that high quality in design, and to enable the intelligent observer to recognise and appreciate it when he meets it in the produc

town council, and the expenses are defrayed from the proceeds of the burgh manure. The inhabitants are principally engaged in the herring-fishery in Loch Fyne, which is said to have produced in some seasons upwards of 20,000 barrels. (Beauties of Scotland, vol. v., p. 437.) The grammarschool is superintended by a teacher, whose salary is 207. The number of scholars during the last 10 years has varied from 25 to 30 annually. The population of the burgh and parish in 1831 was 1117.

Inverary Castle, the principal seat of the Duke of Argyle, is situated near the northern extremity of Loch Fyne. It is a quadrangular building, with a tower at each corner, and a high glazed pavilion rising from the centre of the roof. The stone of which it is constructed, though soft, is very durable, and becomes perfectly black when wetted by a shower. The spacious hall, which is hung with arms and other ornaments, is lighted by a lofty window, and surrounded by a gallery. The other apartments are fitted up in a modern style and with good taste. (Parliamentary Papers; Beauties of Scotland, &c.)

three baillies, and 15 town councillors. In 1832 the estimated value of the burgh property, consisting principally on lands and other heritable property, was 20,8117.; producing an annual revenue of 22367. The annual expenditure at the same period was 20587, and at Michaelmas 1833 the aggregate debt was 10,6147. The town is large and well built; the houses are lofty, and many of them elegant. The streets have, since 1831, been paved with granite and hard sandstone brought from the banks of Loch Ness. Common sewers have been constructed, and the town is well lighted with gas, and supplied with water by means of pipes from the adjacent river. The system of police is also described

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 differ-
ent 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 Inver-
ness 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 mortifica-
tions 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 quan-
tity 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 conglomerate 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 1000l. to 3000l.; 23 from 400l. to 10007.; 33 from 1007. to 4007.; and 14 under 100%. 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 barleymeal 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 21. 108. 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 185,5651. 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,

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.

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 + x* and (x-1) are inverse to one another; for

1 + { √ (x − 1 }2 = x,

We need do no more than name addition and subtraction, multiplication and division, raising of powers and extraction of roots, as pairs of inverse operations.

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 The operation of inversion is the solution of an equation, full operation in 1805, but the whole line of navigation was and vice versa. Let it be required to find the operation innot opened till the latter end of the year 1822. The expense verse to x. Assume pry, and find a in terms of y; of constructing the Caledonian Canal, as it is called, was say ay, then (y) = y, or and are inverse operadefrayed by government. That part of the navigation which tions. Thus if x2 2 x = y, x = 1 ± √y+1, and either of 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 two, 1+x+ 1, or 1 +1, is inverse to x2 - 2x. the original plan contemplated a depth of 20 feet. Loch It thus appears that a function may have more than one Oich is the summit-level of the canal, and its eleva-inverse function, and there are functions which have an intion is 94 feet above the level of the sea on the east finite number; but there is a distinction by which one may coast, at high water and ordinary spring tides. The be separated from all the rest. Let the Greek letters in entire cost was 986,9247., to which must be added a portion this article be all functional symbols, or marks of operations of the annual expenditure since the opening of the naviga- to be performed, and let them come before the subject of tion, which has hitherto exceeded the produce of tonnage operation, the quantity x, or y, &c., in the order in which dues. The tonnage dues on vessels, whether laden or un- they are to be performed. Thus apr denotes the result of laden, is one farthing per ton per mile, and produced in performing the operation upon x, and then the operation 1829 a revenue of 25751.; but the expenditure during the a upon x. Now let z give x=4z, where x is same year amounted to 45737., so that this canal promises an unambiguous operation, and is, generally speaking, to be but an unprofitable speculation. Its chief effect, as ambiguous, or presenting several different forms. Then regards the town of Inverness, has hitherto been the com- and are inverse operations, and z = z, and we might mencement and gradual formation of a direct intercourse suppose at first that x=x; that is to say, we might with the great western marts of Glasgow and Liverpool, and, imagine that destroys as well as that destroys. But through them, with the manufacturing districts with which since is ambiguous, it may be that only one or more of these cities are so closely connected.' (New Statis. Acct.) the forms of will satisfy px, and not all: and that The roads are under the management of the Parliamen- this will be the case with one is obvious, while we can show tary Commissioners for Highland Roads -a body appointed that it cannot happen with more than one. For though the for opening the communication by land about the same time same operation, performed on different functions, may prothat the Caledonian Canal was projected. They are said to duce the same function, yet different operations, performed be maintained in a state of most efficient repair, the expense on the same function, must produce different functions. If being defrayed partly by government and partly by contri- then a and ẞ be different forms of, we have a xx and butions from the county proprietors. The principal rivers ßxx; but we cannot have both ax = x and ẞx are the Spey, Ness, and Beauly; in all of which there are x, where a and B are different, or having absolutely valuable salmon fisheries, more particularly upon the Spey the same form and value in both equations. and Ness.

=

only a function of x, but the same function of other functions of x. Let a x be an inconvertible inverse of x; then a px is not x, let it be x. Then pax being x, pa px is px, or is ox, so that or is the same function of ox which it is of x. Thus in the preceding example x2 - 2 x is the same function of 2 x which it is of x; or

From all the inverses of a function x, then, we separate Education, Schools, &c.-Upon the formation, in the that one, ax, which gives both axx and a&x=x, year 1818, of the Society for Educating the Poor in the and call it the convertible inverse. Its symbol is ̄1, so Highlands,' a central or model school was erected on a large scale in the town of Inverness. This establishment, that 1x means that operation which satisfies both the as might be expected, has already proved highly beneficial to the poor of the town and suburbs, and is upon an average the preceding example 1+(x+1) is the convertible equations 'rx and x=x. [EXPONENT.] In attended by about 300 scholars. The same society supports inverse of x-2x: for 1+ √(x2 - 2 x + 1) = 1 + x · twelve other similar institutions of less extent in the more remote and thinly peopled part of the Highlands, and from. But 1 √x* - 2x + 1 gives 1 · (x − 1) or 2 — x; its first establishment in 1818 to the 30th of September, and we call this an inconvertible inverse. 1834, its total expenditure amounted to 80237. The Rain- Every function which has more than one inverse is not ing 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 407. 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 instruction, which are attended by a very considerable num- We 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 ble inverse to one of its forms, and an inconvertible inverse to Rev. Dr. Bell, the ingenious author of the Madras System all the rest. Thus 1 (x+1), which is an inconvertible of Education, and committed by him to the charge of the inverse to - 2x, is the convertible inverse of (2 — x)2 magistrates of Inverness, who contemplate, we understand, - 2 (2x): for 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.)

x2 - 2 x = (2 − x)3 — 2 (2 – x).

√ +1)=
((2 − x)2 − 2 (2 − x) + 1 ) = 1 −(2−x− 1) = x.

1

The way to make the convertible inverse of a given func

='x we have xx, and these functions will be considered, in the article just cited, as periodic functions of the second order.

The equation 'xx being understood, suppose that between the first and second operations we interpose the operation a, so that we have a $ ̃1x. 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 ẞæ be inverse to each other, then parand ß1 are also inverse to each other: for aßx=x and φαφ pa þ ̃1(†3þ−1) x is þa¿ ̄'ß¿ ̄x, or paß p ̃1x,

or —1x, 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 √(x2+1) and√(x-1). For more detail on this subject see the article Calculus of Functions,' in the Encyclopædia Metropolitana.'

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,

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.