time been bestowed in the construction of an instrument than in the attempt to perfect the needle and mariner's compass. Sir William Snow Harris has, in a form of compass which bears his name, adopted the suggestions of Norman in 1590, as regards the compensating slides on the bar, and also the opinions of Dr. Knight in 1750, with reference to the form of needle; but its chief excellence is in the amount of directive force imparted to it. He also adopts the light form of needle, but not to the extent found in a Chinese instrument. Enough has been said to show that the precise form and arrangement of the mariner's compass has long been a question of public anxiety; and still more must its consideration press upon us when to the new armament of ships and the increased use of iron afloat (as already noticed), must be added the increased amount of iron carried as cargo, and the circumstance of increased speed of ships which brings them more suddenly into danger; these beset navigation with difficulties only to be surmounted by a well-founded confidence in the form, and simplicity in the mode of using a well-made compass, and this can only be attained by careful investigations and unfettered and disinterested conclusions.

It is worthy of remark here that to such an extent has the public mind at times become embarrassed with this consideration, that in 1854 a panic on the Liverpool 'Change nearly excluded for several days the iron ships of the port from freight engagements. It arose from the following circumstance: The late Rev. Dr. Scoresby, with that honesty of purpose and plainness of elucidation for which he was remarkable, informed the merchants of Liverpool of various discoveries as to the causes of local disturbance in iron ships, such discoveries having however a tendency to cast doubt and distrust around that on which hitherto the sailor had relied as his faithful conductor through the pathless oceans of the globe. [LOCAL ATTRACTION.] The insidious workings of the magnetic influence, then for the first time made known to the commercial world, and this too under the sanction of a meeting of the British Association, naturally alarmed the ship-owner, appalled the merchant-captain, and lent its aid towards general confusion. Although the discoveries of Dr. Scoresby are of absorbing interest to the philosopher, and have assisted others in their laboratories, yet to the mariner or the ship-owner these subtle workings of the magnetic current ought to have presented very little real difficulty, as was promptly shown in a published address by Mr. Saxby to the ship-owners of Liverpool, and also by the judicious and timely appeals to them by Mr. Grantham, a very eminent marine engineer of that town. The effect of this on the compass question was, the almost immediate production of such a variety of forms of the instrument as were calculated, it was supposed, to pacify the ship-owners; but it actually left the ship-captain burdened and bewildered with novelties and perplexities which even now, in a great measure, render the correction and use of the compass, notwithstanding its importance, the least satisfactory, the most anxious, and the most tiresome of all his work at sea. This competition, however painful in its early operation, had, in a national point of view, a salutary effect. It loudly evoked the hidden talent of officers in the mercantile marine. It aroused their energies, and demanded of them something beyond attention to the mere routine in which they had been trained. Ship-captains soon found among their numbers many men of sufficient talent to grapple with the compass question, and that large and respectable body, relying on themselves rather than on men of reputed science, began to judge for themselves as to the merits of the various toys placed into their binnacles by credulous owners; and many an absurd, though presumptuous and specious form of compass has already been by them consigned to oblivion; many ingeniously contrived instruments, good in themselves but totally unfit for sailors' use, have gradually sunk or are yet gliding visibly into disrepute. With that promptitude, however, which characterises Liverpool merchants, soon after the announcement made by Dr. Scoresby, a Compass Committee, seconded by the authorities at the Board of Trade, was formed, and a vast collection of facts speedily poured in from those who were interested, and measures were taken to assist navigators, which will be clearly and more properly detailed under the head of LOCAL ATTRACTION in this work. But nothing resulted from their really praiseworthy and indefatigable investigations, calculated to modify the generally existing form of mariner's compass: nothing new appears to have been suggested by that body indicative of real improvement in the instrument.

Although we must not in this article enter far into the question of local attraction, yet as its causes have led to some modifications in the form of compasses to facilitate corrections for deviation, &c., it is necessary to notice that the Astronomer Royal, having long turned his powerful mind to the question, announced to the world a theory of local attraction which involves the necessity for soft iron as a part of the compass-correcting arrangement. No greater proof of the difficulties which surround the mariner's compass as a nautical instrument can be adduced than the fact, that, although now some few years have passed since Professor Airy first gave to the world his really simple and elegant theory for correcting by magnets and soft iron, his opinion remains unseconded and unadopted by the Admiralty, by whom correcting magnets are rejected altogether, from extreme and irreprehensible, but possibly overstrained, prudence. It is not improbable that a number of the fallacious methods of correction of the compass produced of late, have so deluged the authorities at the Admiralty

with suggestions thereupon, as to somewhat blunt the intelligence, and thereby obscure the truth, even when offered by the AstronomerRoyal himself. It must also be remembered, and indeed it is a fact well-known to the writer, that many merchant-captains who have had their compasses corrected in port by the use of fixed magnets, have found such error and inconvenience arising therefrom (on account of subsequent changes in the magnetic condition of the ship), that rather than endure the annoying discrepancies and perplexities consequent on their use, they have pitched their correcting magnets overboard altogether; preferring to encounter the magnetic dangers of the voyage with a single well-made compass which they did understand, to tampering with changeable and vacillating agents and appliances which were above their comprehension. Such may account for the authorities at the Admiralty rejecting the use of correcting magnets altogether.

Taking even Professor Airy's theory of moveable adjusting magnets and soft iron as an instance, how few merchant-captains, unaided by a mechanically convenient correcting apparatus, could follow a learned discussion on so intricate a science as that of magnetism (however valuable in itself); much less could they use the mathematical formulæ necessary for its application to the mariner's compass. It remains to state that a practised compass-maker, a Mr. John Gray, of Liverpool, an optician of talent well known in all the principal ports of Europe, after study and experiment on an extended scale in which he believed he could corroborate physically the theoretical assumptions of Professor Airy, contrived a compass arrangement which, apart from all invidious selection, and from a sense of public duty, finds an illustration in this article. Several other forms for correction of the compass-needle exist, and each has its advocates, but the following sketches of Gray's compass arrangement give an idea of the form in which the Astronomer-Royal's suggestions are being carried out.

B,

Outer bowl, containing finid, in which, в 2, inner bowl is floated; R and R 2, Rims of outer and inner bowls; v s, Springs to keep the bowls in central position with each other; T s, Tangential screws to adjust the bowls to their centres; G and G 2, Guides to prevent rotatory motion; L, Lubber's point; D, Elastic discs; s, Spiral spring; c, Chain boxes for soft iron; a 3, Gimble.

The above has, moreover, an interest from being the form of apparatus used on board Her Majesty's yachts; and without compromising that

exist in our system. In such a cause, it behoves every man of science who would assist the navigator to cast aside all prejudices and mere "customs," and, taking the simple facts of the case, attempt a total revision of the subject of compass correction.

But what of the check referred to, as deduced from celestial observation? The question of correction must be viewed under two aspects, namely, the accuracy and labour in the means employed. As a question of spherical trigonometry, its accuracy is mathematically sufficient; and, as regards labour, there are two ways of working, namely, by logarithmic calculation and by construction. The calculation of an azimuth is the resolution of a spheric triangle, in which certain things are given to find others: as in the following example, in which it was possible to use the horizon.

Suppose a ship to be in latitude 51° 30' N., when the altitude of the sun's centre was 40° 25', the sun's declination at the same moment being 20° 2' N.: required the sun's true bearing and the error of the compass, the bearing of the sun by compass being S. 79° 39' W.:

North

Error of compass

Such are the calculations in each case, and they are shortened when altitude and time are both known. Better even would it be to use in this manner a few extra logarithms daily, than to depend on a correctingcard. But there is another and more simple method, not generally known, of solving a spheric triangle with sufficient accuracy for an azimuth, where the nearest degree is enough, because one cannot steer a ship to within less than a degree or two: it is by construction of the spheric triangle. This process, however, would have its inconveniences, although it requires only a plane scale and a pair of dividers or compasses. But these inconveniences have of late been totally obviated by the invention of the spherograph [SPHEROGRAPH], in which, having any three elements of a spheric triangle, the others are found without any calculation, and in a very few seconds; indeed, the Astronomer-Royal has given his written opinion that "for the special purpose of determining azimuths to correct a compass, he thinks the spherograph is excellent."

In anticipation of the word SPHEROGRAPH, a sketch of the instrument as it appears when finding an azimuth will in this place be sufficiently illustrative.

In the annexed figure the sun is represented as being just on the horizon, the dark lines of the drawing represent the upper sphere, and the dotted ones those of the under sphere as seen through the upper transparent one, both spheres being moveable on the centre c.

the horizon), and the oblique circle Pos to represent the hour circle for 7 P.M., and the small circle Doм to represent the sun in its distance from the equator (EQ), or, as it is called, the declination. For all nautical purposes the line DOм may be called the sun's path in the heavens for that day. Now, the part of the circle R must have been moved on its centre o so as to place it 50° (the required latitude) from P: on the under sphere no other movement is required: but we see at a glance that the sun would be at M at midnight, at o when rising or setting, and at D when on its meridian. Hence, the degrees being all printed on the spherograph so as to enable all distances to be read off, no further measure or construction could be required, and we should in this instance find that co on the horizon would measure the rising amplitude and OR the rising azimuth; and c being the east and R the north part of the horizon of the instrument, the sun would rise at about N.E. by E. Suppose it were required to correct the compass at any time of the day, say at 9 A.M., I should select the 9 A.M. hour circle Ps, as drawn on the under sphere of the spherograph, and notice where it crosses the parallel of declination DM at ; and any vertical circle (suppose zN) which, passing through the intersection, cuts the horizon (as at x), would show the bearing in azimuth as measured at HX; and as H is at the south part of the horizon and c is at the east, cx would be very nearly east by south: if the compass showed by it that the sun was at the same time (for instance) E.S.E., the compass would have an error of one point.

It seems then that a possibility exists of totally avoiding accidents dependent on compass errors, and by a means sanctioned and approved by the Astronomer Royal; and although changes in magnetic condition cannot be foreseen and prevented, we in reality seem to be able, by merely turning a transparent card on its centre, and at any time, or at any part of the world, and by a process which occupies about ten seconds of time, when any heavenly body is visible, and without requiring any observation for altitude, to place an effectual check upon

a compass.

In order therefore to put the question in a plain and available form for mariners, the following is proposed as a system applicable at all times to any compass, and on board any ship, be her magnetic condition whatsoever it may : and as this method of ascertaining a true azimuth from celestial observation is totally independent of the horizon, it is available in a few seconds whenever any heavenly body is visible, even in hazy weather, and is therefore much more convenient and expeditious, as it must be more accurate, than any swinging of a

ship:

1. As a mariner always knows his latitude and declination to the nearest degree, and his apparent time to the nearest minute or two, let him, when desirous of merely checking his compass, find at once the sun's true azimuth [BEARING] by the spherograph (or by construction or calculation, at pleasure), and compare the result with his compass bearing of the sun's centre. If his ship be at anchor, and he wishes to examine into her general local attraction, let him, while she is swinging with the tide, take the bearings of the ship's head as she comes to the several points of the compass, noting against each observation the apparent time at which it was taken: then let him seek against each of those times in the spherograph the corresponding true azimuth of the sun (the whole is done by one movement of the upper card of the spherograph on its centre), and by comparison the whole condition of the ship for that day or period is shown, and that too without the usual labour and expense and detention of the vessel during compass correction. As many as 300

bearings have been taken by the late secretary of the Compass Committee at Liverpool, while a ship has been swinging with the tide even in so quick and strong a river as the Mersey. 2. Either register the error so found in one of the admiralty forms of graphic delineation (Sir A. Smith's), or adjust the shifting magnets and soft iron by the astronomer royal's method, by the assistance of Gray's or any other apparatus for effecting this object. [COMPASS, THE MARINER'S.]

3. Let the ship occasionally yaw a few points from her course at least once a day, using the spherograph, and in case of suspected bad weather, up to the last moment at which a heavenly body is visible.

By following the above plan the compass need no longer be a source of anxiety to an officer of a ship; for instead of his having, as at present, to depend on an erroneous system of adjustment made in port perhaps many months before, or by the working of an azimuth, for which he cannot always get an altitude, he can dispense with both these methods, and avoid all calculations or complexities whatsoever, whenever any heavenly body is visible. If by night, he would use the star's distance in time from the meridian of the place as if it were in the spherograph apparent time.

From the extreme simplicity, infallibility, and rapidity of the above method, it may be suggested that, in passenger over sea steamers, the compass ought to be checked once in every watch; for this purpose a common compass, fitted with Captain Robertson's patent "deviation detector" [BEARING], would, with the spherograph, be all

sufficient.

by the outer rim of the card, on which the degrees are generally marked. The direction in which an object lies is called its bearing. [BEARING.] COMPASSES. This term we suppose to be synonymous with compassers, instruments by which we compass or go round a space. We shall here only give such a general notion of differenf kinds of constructions as will perhaps suggest the most convenient for any par ticular purpose.

COMPASS, NOTATION OF. The notation of the mariner's compass is very simple. If we divide a circle into four equal parts, each of the points which separate those parts may be called cardinal. Pliny called one of these the cardo mundi, or the pole of the world, meaning the north pole. Another ancient author speaks of the cardo cali, or the pole of the heavens, meaning also the north pole; while Quintilian speaks of the four as the quatuor cardines mundi. Hence, by common consent, the north, east, south, and west are called "cardinal points." The formation of the compass card will be easily understood by the following:

1. Common Compasses, or Dividers.-These are simply two pointed legs on a common pivot, for transferring distances. For drawing a circle the lower end of one of the legs is removed, and its place supplied by a holder for a pencil, or by a steel pen.

2. Hair Compasses.--One of the legs has a part attached to the upper part by a spring, so that by means of a screw a very small motion may be given to the lower end. It is convenient for very accurate dividing, but must be used with care.

3. Triangular Compasses.-These have three legs and two pivots, so that the three points of a triangle can be at once transferred. This is useful only in rough work, as the instrument is difficult to handle.

4. Proportional Compasses.-These consist of two dividing compasses with a common pivot, which, when open, present vertically opposite other are in the same proportion as the legs of one to the legs of the angles; consequently, the intervals between the points of one and the other. The pivot is a clamping screw, which can be transferred along the interval between the pairs of points, and a scale points out how to adjust the instrument so as to alter any line, or surface, or solid, in a given proportion. These compasses sometimes have an apparatus for slight adjustment; but on the whole we consider it as an instrument for rough work.

cular to which, with clamping screws, slide a point and a pencil. The 5. Beam Compasses.-This instrument is a cylindrical bar, perpendiuse of it is to describe large circles, or to measure large distances, the It is a safe and sure construction. common compasses being very liable to slip when opened very wide.

6. There is a method of describing a small arc of a very large circle, as follows: An elastic rod of metal is furnished with a rigid bar, on which it can be drawn up by screws, so that the rod shall form an arc, the chord of which is a part of the bar. This may be adjusted so as to pass through three given points nearly in the same straight line, and though the curve then described by guiding the point of a pencil along the rod be not exactly an arc of a circle, yet, for all small flexures, it will come sufficiently near for practical purposes.

7. Caliper Compasses, or callipers, are compasses intended to measure the calibre or diameter of round bodies, and are formed with curved legs, knobbed instead of pointed. Being opened until the body to be measured can only just pass through them, the distance between the two internal extremities of the knobs is of course the diameter of the body.

Let N.E.S.W. represent the cardinal points. The point midway between them is formed by combining the letters; thus, taking the quadrant or quarter of a circle which lies between N. and E., the intermediate point will be called N.E. If we halve this distance, N. and N.E., we call the middle point N.N.E. If we halve now the distance, N. and N.N.E., we call it N. b. E. (or north by east, or north towards east, for any further combining of letters would be inconvenient). In like manner we divide N.E. and E., and the intermediate point will of course be E.N.E. (the nearest cardinal point always stands first when we combine the letters); and again, halfway between E.N.E. and E. would be E. b. N. (or east towards north). It need only to be remembered that the cardinal points, and the midway points between them, such as N.E., S.E., N.W., S. W., always have the word "by" or "b." in the points next to them. The other three quadrants are formed in precisely a similar manner. We thus find the circle divided into 32 parts or points; and as the whole circumference of a circle is divided into 360 degrees, 360 divided by 32 will give 11° 15', or 114 degrees, as the angle which each point makes at the centre of the compass.

When using a compass card, the observer should always consider himself as at the centre of it, and not outside the circle; for the centre of the compass card represents the point of the earth on which he is standing, and the visible horizon may be considered to be represented

Many other species of compasses have been constructed, but the above are the principal ones in common use. [ELLIPTIC COMPASSES.] COMPLEMENT, that magnitude which, with another, makes up a given magnitude. This is the general meaning of the term; but the most usual specific uses are as follows: Complements of the parallelograms about the diagonal of a parallelogram: through a point in the diagonal draw parallels to the sides; the whole is then divided into two parallelograms on the diagonal, and two which only touch the diagonal at one angle. The latter pair are called by Euclid complements to the former.

The complement of an arc or angle is the arc or angle by which it falls short of a quadrant or a right angle.

The complement of a logarithm is the number by which a logarithm falls short of 10: thus comp. log. 2 is 1030103 or 9'69897.

The arithmetical complement of a number is the number by which it falls short of the next higher decimal denomination. Thus, ar. co. 936 is 1000 - 936, or 64; arith. comp. of 83 is 100 - 83, or 17. Beginning from the left, subtract every figure from 9, up to the last significant figure, which subtract from 10.

For the complement of life, see DE MOIVRE'S HYPOTHESIS.
COMPLEMENTARY COLOURS. [LIGHT.]
COMPLUTENSIAN POLYGLOTT. [BIBLE.]
COMPOS, or NON COMPOS, MENTIS. [INSANITY.]
COMPOSITE ORDER. [COLUMN.]

COMPOSITION. In the gradual progress of mathematical language, this word has acquired a general meaning, as follows: Any one magnitude is said to be compounded of two others, when it produces the same effect as the other two put together. For instance, if we increase a length in the proportion of 3 to 7, and then increase the result in the proportion of 2 to 5, the original line is increased in the proportion of 3 x 2 to 7 x 5, or of 6 to 35. Hence the proportion of 6 to 35 is said to be the proportion compounded of (out of) the proportions of 3 to 7 and 2 to 5.

The effects of which it is in our power to form a distinct conception are of two kinds: 1. Those in which there are only two kinds imaginable, and those two diametrically opposite, with one neutral inter mediate state. 2. Those in which the diametrically opposites have an infinite number of intermediate gradations. Loss or gain of money is an instance of the first; change of direction of the second. If, at the rate of an inch to a shilling, gains were measured northward from a

given point, and losses southward, we could immediately make it a necessary consequence that the balance, if any, is represented by a line northward or southward, according as it is for or against. But draw a line eastward, and it will readily be admitted that such line will not present itself in any necessary connection with a sum lost or gained, or neither lost nor gained. For if the latter, why should a line eastward be preferred to a line westward, or in any other direction?

An immense number of modes of composition will readily suggest themselves, in which addition and subtraction are the processes by which composition takes place. If I go three miles northward, and then two miles farther, I go in all 3 + 2, or 5 miles northward. Other modes, as in the instance first given, will suggest themselves, in which multiplication and division are the compounding processes, and so on ad infinitum. These are all cases in which magnitude only is concerned; but whenever we have both magnitude and direction, it is plain that we have now both magnitude and direction to consider in the effect. If I go a mile northward and then a mile eastward, the whole effect, as to direction, will be, that I go to the north-east; as to magnitude, that I go not two miles, but only 2 miles, or 1.414 miles very nearly. Here is an instance in which the components are represented in magnitude and direction by two sides of a triangle, while the total effect is similarly represented by the third side. In the article CENTRE will be found various instances in which the meaning of that term implies the point at which a single action must take place, which will produce the same effect as a number of different actions produced on a number of points.

In mechanics, we have to consider the combined effects of different velocities, pressures, momenta, rotations, &c., communicated at the same moment of time to the same body. In all, the law of composition

is found to be as follows:

In what sense soever the actions at P can be represented in magnitude and direction by P A and PB, in that same sense can the joint effect be represented by PC, the diagonal of the parallelogram, in both magnitude and direction: or P A and AC being the actions (A C being equal to P B in magnitude and direction), P c, the third side of the triangle, is the united action. Thus, if at the same instant we communicate motion to P in the directions PA and PB, with velocities PA and PB per second, we thereby merely communicate to P a velocity PC per second, in the direction P C. The same holds of momenta and pressures; and even if we give P two separate rotations, which would separately carry it round the axes PA and P B in angles per second which are in the same proportion as PA and P B, the joint effect is a rotation round the axis PC with an angle per second which is to the angle of PA (or P B) as P C is to P A (or P B).

We have here not to prove these things, but only to illustrate the word composition. But this we must remark, that our preconceived notions will never allow us to say that A is the effect of P and Q, and B of R and s, unless the application of P, Q, R, and 8 together will be the same in effect as that of A and B together. We shall show that this necessary condition of our notions of cause and effect is preserved in the method of composition just described. Let P X, B Y, and cz be parallel to each other; then, if our law of composition be general, PB is the effect of PX and PY. Therefore PX, P Y, and P A should be together equivalent to PC. But A z is equal to PY, and P Z is therefore equivalent to PA and PY. Therefore A c should be the effect of PZ and PX, which we immediately see it is, being as much the diagonal of

PX CZ as it is of P B C A.

As another, and a very curious instance of composition, we shall notice the following: Suppose x and y are to be measured, and both are subject to error, every error entailing loss in proportion to its magnitude, and causing equal loss, whether it be an error of excess or an error of defect. Suppose also that the errors are of such a kind that the average of any number of measurement is more probably right than any other. Let a and b be the sums which it would be equitable to pay for insuring x and y, that is, which should be given to any one who would agree to bear the loss on x and y separately. The sum which should then be given to one who would bear the total loss arising from the possible error in x + y is not a + b, as might at first appear, but a+b2, or the hypothenuse of a right-angled triangle of which a and b are as the sides.

Our limits will not allow us to enlarge on this subject; we shall add the two following remarks.

1. The fact of the law of composition being the same both for velocities and pressures, has caused many writers on mechanics to confound the two, as if the one proved the other, which is neither true, nor even very probable. And other writers on mechanics, while proving this general law, that actions which can be represented by the two sides of a triangle produce an action which can be represented by

ARTS AND SCI. DIV. VOL. III.

the third side, have restricted the proposition, and seem to imagine that what they prove is true of forces only. This, with great deference to such a writer, we conceive M. Poisson to have done in the wellknown proof at the beginning of his mechanics, a work which we may take this opportunity of saying, we hold in higher estimation than any other elementary mathematico-physical work whatsoever.

2. The difficulties of negative quantities in algebra arose from a want of generality, which gave rise to the attempt to express composition by addition only or by subtraction only, where either addition or subtraction might be requisite; and the difficulties of impossible quantities arose out of a similar deficiency, bearing the most complete analogy to trying to compound in magnitude only, in cases where both diversity of magnitude and of direction should have been considered. COMPOST. [MANURE.]

COMPOUND, that which results from composition. [COMPOSITION.]

COMPOUND ADDITION. [ADDITION.]
COMPOUND INTEREST. [INTEREST.]

COMPOUND QUANTITIES [ARITHMETIC], quantities in which more than one unit is employed, as in 2 pounds, 3 shillings, and 6 pence: 2 miles, 3 yards, and 4 inches.

COMPOUND RADICALS. A term applied in chemistry to those combinations of elements which act towards oxygen, hydrogen, and acids, as simple elements. [ORGANIC RADICALS.] COMPOUND RATIO. COMPOSITION; RATIO.] COMPOUNDING OF A FELONY. [FELONY.] COMPRESSIBILITY OF WATER. [ELASTICITY.] derived from the canon law, of permitting persons accused of certain COMPURGATOR. In the middle ages a practice prevailed, crimes to clear themselves by purgation. In these cases the accused party formally swore to his innocence, and, in corroboration of his oath, twelve other persons, who knew him, swore that they believed in their consciences that he stated the truth. These twelve persons were called compurgators. (Ducange, Juramentum.') This proceeding appears to have existed among the Saxons, and, in process of time, it came into use in England in civil cases of simple contract debts. [WAGER OF LAW.] The ceremony of canonical purgation of clerksconvict, which was nothing more than the formal oath of the party accused, and the oaths of his twelve compurgators, continued in England until it was abolished by the stat. 18 Eliz. c. 7. [BENEFIT OF CLERGY.] (Blackst. 'Comm., Dr. Kerr's edit., vol. iii., 364; iv., 434.) COMPUTATION. We need not tell those who are acquainted with the existing treatises on arithmetic, that in no one instance do they pretend to give any mode of forming good habits of computation. The beginner, after receiving instructions as to what is to be done in the several great rules of arithmetic, is allowed to manage the details

as he can.

That we

The mere mechanical art of computation, apart from arithmetical reasoning or application to subjects of interest, is no very lofty exercise of the mind. A wonderful degree of proficiency in it can be attained by many who find connected reasoning almost an impossibility; and on the other hand, some of the first among mathematical discoverers have hardly arrived at more than the expertness of an ordinary schoolboy. It is one of those arts among many which are accessible to all who begin with a determination to conquer difficulties, and a power of arriving at methodical habits: no person who, after beginning in the right way, is obliged to confess a total failure, has any ground to suppose that he could master a common manual art: he may be a genius, but nothing could make him a weaver. may not frighten any one of the thousands who are miserable computers after going through years of school discipline, and whose minds are too well made to allow them to flatter themselves that they were above it, it is but fair to say that very few are allowed to begin in the right way. Every merely mechanical business must be learned by a sufficient repetition of the most purely elementary steps. discipline of the mind may be taken up at the wrong place, and still be a discipline, though not so perfect as it might have been. But in what is merely art, nothing can compensate for the want of habit of operation duly learned at the proper time. Now computation is only an art its elements are a small number of acts of memory: its details consist in a still smaller number of operations, each of which, by itself, is of the utmost simplicity.

A

Many readers will suppose us, in speaking of the elementary rules of arithmetic, to mean addition, subtraction, multiplication, and division, as given in the books: but we should as soon think of saying that the elementary operations of a journeyman tailor's business are the making of coats, waistcoats, and trowsers. The rules just named are the perfection of computation, not its commencement: he who can do them all with ease and accuracy is a calculator. The fundamental operations of which we speak are to those elaborate processes just what threading a needle and drawing a stitch are to the making of a coat. We can carry the comparison still further: and its justice is not accidental, but the necessary consequence of the resemblance of all mechanical operations. We do not say that a workman who is capable of joining two pieces of cloth together with strength and neatness is a finished tailor: he cannot therefore choose cloth, cut out

I