495 



UNIFORM. 



UNIT. 



ever at last discovered that it was fond of rarities, and particularly 

 attached to chaste persons ; BO they took the field with a virgin, who 

 was placed in the unsuspecting admirer's way. When the Unicorn 

 spied her, he approached with all reverence, couched beside her, and, 

 laying his head in her lap, fell asleep. The treacherous virgin then 

 gave a signal, and the hunters made in and captured the simple beast. 



Modern zoologists, disgusted, as they well may be, with fables of 

 which we have only given a specimen or two, disbelieve generally the 

 existence of the Unicorn, such at least as we have above referred to ; 

 but the result of M. Guettard's dissertation is an opinion that some 

 terrestrial animal bearing a horn on the anterior part of its head exists 

 besides the Rhinoceros. The nearest approach to a horn in the middle 

 of the forehead of any terrestrial mammiferous animal known to us is 

 the bony protuberance on the forehead of the Giraffe [GIBAFFE, in 

 NAT. HIST. Div.] ; and, though it would be presumptuous to deny the 

 existence of a one-horned quadruped other than the Rhinoceros, it may 

 be safely stated that the insertion of a long and solid horn in the living 

 forehead of a horse-like or deer-like cranium is as near an impossibility 

 as anything can be. 



The " Monoceros home " in Tradescant's collection was probably 

 that which ordinarily has passed for the horn of the Unicom, namely, 

 the tooth of a Narwhal. Old legends assert that the Unicorn, when he 

 goes to drink, first dips his horn in the water to purify it, and that 

 other beasts delay to quench their thirst till the Unicorn has thus 

 sweetened the water. The Narwhal's tooth makes a capital twisted 

 Unicorn's horn, as represented in the old figures. That in the reposi- 

 tory of St. Denis, at Paris, was presented by Thevet, and was declared 

 to have been given to him by the king of Monomotnpa, who took him 

 out to hunt unicorns, which are frequent in that country. Some have 

 thought that this horn is a carved elephant's tooth. There is one at 

 urg some seven or eight feet in length, and there are several in 

 Venice. 



Great medical virtues were attributed to the go-called horn, and the 

 price it once bore outdoes everything except the Tulipomania. A Floren- 

 tine physician has recorded that a pound of it (sixteen ounces) wan sold 

 in the shops for fifteen hundred and thirty-six crowns, when the same 

 weight in gold would only have brought one hundred and forty-eight 

 crowns. 



The Unicorn is a national symbol with us, for it is one of the sup- 

 porters of the royal arms of Great Britain, in that posture termed by 

 heralds " saillant." It was introduced as one of the supporters of the 

 English arms by James I., who having as king of Scotland borne two 

 unicorns, coupled one of them with the English lion on his accession 

 to the throne of England. 



UNIFORM. Though this word mean nothing more than " of one 

 form," it has a signification in mathematics which might be better 

 rendered by " of one value " or " of one degree," when we speak to the 

 mathematical proficient. But it is a convenience, though only an 

 accidental one, that the word does not imply the idea of value abso- 

 lutely ; a circumstance which may serve us to elucidate a point of great 

 importance in the differential calculus. The commencement is made 

 in the present article : the continuation will follow in VELOCITY. 



In order to understand any application of mathematics, whether to 

 ipace or matter, it is necessary that a perfect mathematical conception 

 should be formed of the quality of space or matter which is to come 

 under consideration. By a perfect mathematical conception, we mean 

 that it must be distinctly seen, first, that the object under considera- 

 tion is of the nature of magnitude ; secondly, that it is of a measurable 

 kind, that is, is capable of being measured, and can actually have a 

 mode of measuring it assigned. Why do so many persons talk and 

 write vaguely about force, velocity, density, acceleration, &c. ! Simply 

 because they are only conversant with the first consideration, and have 

 no precision in their ideas of the second : they feel that they are 

 speaking of magnitudes, of things which they know may be more or 

 "it they have not that familiarity with the precise way of ascer- 

 taining the haw much more or the how much lest, without which deduc- 

 tion cannot be made intelligible. 



Now we say that in every instance in which measurement is shown 

 to be attainable, there is a notion of uniformity which precedes or 

 ought to precede that of mensurability ; and that emphatic mention 

 of this circumstance, and full development of its truth and meaning, 

 ought to be the preliminary step to actual measurement. Moreover, 

 we say that, inasmuch as this idea of uniformity is to be gained pre- 

 viously to that of measurement, we must forego the notion of " uni- 

 form " and " of one value" being convertible terms, and illustrate the 

 word by considerations independent of value ; for this last term implies 

 measurement, as is easily seen. 



If we were to take velocity as our instance, most readers would be 

 able to appeal to ideas of measurement and value established in their 

 mirirla, whether vaguely or precisely : we therefore prefer to choose 

 curvature, a term which will be quite new as meaning a measurable 

 mignitadt to all except those who have more than an elementary 

 knowledge of mathematics. Curvature is, as the name imports, the 

 bending, the gr;< ing, which distinguishes a curve from a 



traight line. It in a magnitude, that is, it allows of the application of 

 the idea of more and less : one curve may bend more than another, or 

 more in one place than in another. So much every one can be sure 

 of at the first announcement : the next step would be to imagine it 



possible, that one curve might, say at and about a point A, bend 

 exactly twice as much as another at and about a point B. But here 

 the ordinary reader can only imagine a possibility : no distinct criterion 

 will at once present itself for determining what proportion the bendiugs 

 or curvatures of two curves are to be stated as having to one another 

 at two given points. If two tangents be drawn at the two given points, 

 it is obvious that, according as the curve bends more or less, there will 

 be more or less deflection from the tangent. Thus the curve A P, at 

 the point A, has as much curvature as A Q, or more ; certainly not less. 

 Now as in other cases, if we measure curvature, it must be by curva- 

 ture, as length by length, weight by weight, &c. : and as a preliminary, 

 it will be desirable to have that curve which has everywhere the same 



curvature. This curve is obviously a circle, which ia throughout its 

 circumference bent in exactly the same manner. Those who cannot 

 imagine how curvatures are to be measured can always see this much, 

 that a true mode of measurement will give the same result to what- 

 ever point of a given circle it may be applied. A method of deter- 

 mining value must be false which gives at one point of the same circle 

 a greater curvature than at another. Here we say that any one may 

 see that a notion of uniformity has a useful existence previously to 

 that of any mode of comparing the values of different cases of this 

 uniformity. The circle A may have a radius twice as large as that of 

 B : are we then to say that the curvature of B is double that of A ? 

 That the smaller circle bends most is certain ; whence it is equally 

 certain that curvature or bending is a magnitude : it has its more and 

 less. Again, it is obvious that the circle B has the same curvature in 

 all its parts, and that the circle A has the same ; though the parts of A 

 have a curvature which is not the same as that of the parts of B. 

 Hence it is certain that uniformity of curvature is perfectly conceivable. 

 Now what we have to enforce is, that all this takes place in the mind, 

 before any mode can be given of answering the question how much 

 the curvature of B exceeds that of A. The greater the radius the less 

 the curvature, and A has twice as great a radius as B. If it be proper 

 to say [VARIATION] that the curvature varies inversely as the radius, 

 then B is twice as much curved as A ; but if it be proper to say that 

 the curvature varies inversely as the square of the radius, then that of 

 B is four times as great as that of A. Here the object of this article 

 ends, and we have referred to VELOCITY the manner of making the next 

 step. At the risk of undue repetition, we state again, that a perfect 

 idea of a magnitude, as a magnitude, and of its uniformity, or total 

 absence of change of value, may exist in cases in which the accurate 

 comparison of values, or measurement, is not attained, and may even 

 exist in a mind which has not the means of conceiving the possibility 

 of such comparison or measurement being accurately made. 



UNIGENITU.S, BULL. [BULLS, PAPAL.] 



U'NISON, in Music, is a sound which is exactly the same as another, 

 in regard to pitch that is, to acuteness or gravity. 



UNIT or UNITY, the name given to that magnitude which is to 

 be considered or reckoned as one, when other magnitudes of the satno 

 kind are to be measured. It is not itself one, but is the magnitude 

 which one or 1 shall stand for in calculation : it is a length, or a weight, 

 or' a time, as the case may be, while 1 is only a numerical symbol. This 

 symbol 1 represents the abstract conception of singleness, as distin- 

 guished from multitude, and is the unit of abstract arithmetic; but all 

 concrete quantities must have units of their own kind. 



Unity, says Euclid (book vii., def. 1), is that according to which each 

 of existing things is called one : Moras ton, Kofl' V a0Toj' -rav urruiv 

 'in \tyercu. And, allowing somewhat for idiom, it would not be easy 

 to mend this definition. Anything may be unity, for things of ita 

 own kind. 



The common division of units into abstract and concrete is merely 

 the distinction between the unit of numeration and that of measure- 

 ment : the former implying that reckoning or computation is to be 

 performed, without specific reference to any particular object of 

 reckoning ; the latter, that some certain unit of length, of capacity, or 

 whatever it may be, is to be signified by 1, On this point the 

 learner must take pains to see, that of all the fundamental operations 

 of arithmetic, three are wholly independent of this distinction, which 

 cannot be said of the fourtli. Addition, subtraction, and division can 

 be physically performed, and without reference to units : two lines 

 may be put together into one line, a line may be cut off from another, 

 or a line may be carried along another time after time, until it is seen 

 how many times the greater contains the less. But rnultiplicil ion 

 requires that number or magnitude should be t.-iken a number of timet, 

 and the idea of multiplying a magnitude by a magnitude involves an 

 al)surdity. [MULTIPLICATION ; RECTANGLE.] Nevertheless some enter- 

 prising writers on arithmetic profess to multiply magnitude by magni- 

 tude; and, to make their doings more striking, they often choose 

 for their instance to multiply Ml. 19s. U^d. by 991. 19s. lljrf. To 

 take a humbler case, let us examine the product of 5 shillings and 

 3 shillings : beginners educated in the common system of -arithmetic are 

 generally loth to part with the idea that thU must bo 15 shillings. 



