# CIRCULAR

, appertaining to a circle; as a circular form, circular motion, &c.

Circular Lines, a name given by some authors to such straight lines as are divided by means of the divisions made in the arch of a circle. Such as the Sines, Tangents, Secants, &c.

Circular Numbers, are such as have their powers ending in the roots themselves. As the number 5, whose square is 25, and its cube 125, &c.

Napier's Circular Parts, are five parts of a rightangled or a quadrantal spherical triangle; they are the two legs, the complement of the hypothenuse, and the complements of the two oblique angles.

Concerning these circular parts, Napier gave a general rule in his Logarithmorum Canonis Descriptio, which is this; “The rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts, and to the rectangle under the cosines of the opposite parts. The right angle or quadrantal side being neglected, the two sides and the complements of the other three natural parts are called the circular parts; as they follow each other as it were in a circular order. Of these, any one being fixed upon as the middle part, those next it are the adjacent, and those farthest from it the opposite parts.” Lord Buchan's Life of Napier, pa. 98.

This rule contains within itself all the particular rules for the solution of right-angled spherical triangles, and they were thus brought into one general comprehensive theorem, for the sake of the memory; as thus, by charging the memory with this one rule alone: All the cases of right-angled spherical triangles may be resolved, and those of oblique ones also, by letting fall a perpendicular, excepting the two cases in which there are given either the three sides, or the three angles.— And for these a similar expedient has been devised by Lord Buchan and Dr. Minto. “M. Pingre, in the Memoires de Mathematique et de Physique for the year 1756, reduces the solution of all the cases of spherical triangles to four analogies. These four analogies are in fact, under another form, Napier's rule of the circular parts, and his second or fundamental theorem, with its application to the supplemental triangle. Although it would be no difficult matter to get by heart the four analogies of M. Pingre, yet there are few persons blessed with a memory capable of retaining them for any considerable time. For this reason, the rule for the circular parts ought to be kept under its present form. If the reader attends to the circumstance of the second letters of the words tangents and cosines being the same with the first of the words adjacent and opposite, he will sind it almost impossible to forget the rule. And the rule for the solution of the two cases of spherical triangles, for which the former of itself is insufficient, may be thus expressed: Of the circular parts of an oblique spherical triangle, the rectangle under the tangents of half the sum and half the difference of the segments at the middle part (formed by a perpendicular drawn from an angle to the opposite side), is equal to the rectangle under the tangents of half the sum and half the difference of the opposite parts. By the circular parts of an oblique spherical triangle are meant its three sides and the supplements of its three angles. Any of these six being assumed as a middle part, the opposite parts are those two of the same denomination with it, that is, if the middle part is one of the sides, the opposite parts are the other two, and, if the middle part is the supplement of one of the angles, the opposite parts are the supplements of the other two.—Since every plane triangle may be considered as described on the surface of a sphere of an infinite radius, these two rules may be applied to plane triangles, provided the middle part be restricted to a side.

“Thus it appears that two simple rules suffice for the solution of all the possible cases of plane and spherical triangles. These rules, from their neatness and the manner in which they are expressed, cannot fail of engraving themselves deeply on the memory of every one who is a little versed in trigonometry. It is a circumstance worthy of notice, that a person of a very weak memory may carry the whole art of trigonometry in his head.” Napier's Life, pa. 102.

Circular Sailing, is that performed in the arch of a great circle.—It is chiefly on account of the shortest distance that this method of sailing has been proposed; and for the most part it is advantageous for a ship to reach her port by the shortest course.

As the solutions of the cases in Mercator's sailing are performed by plane triangles; so the cases in greatcircle sailing are resolved by the solution of spherical triangles. But, after all, the several cases in this kind of sailing serve rather for exercises in the solution of spherical triangles, than for any real use towards the navigating of a ship.

Circular Spots are made on pieces of metal by large electrical explosions. See Philos. Trans. vol. 58, pa. 68; also Priestley's Hist. of Electricity, vol. 2, sect. 9, edit. 8vo.

These beautiful spots, produced by the moderate charge of a large battery, discharged between two smooth surfaces of metals, or semi-metals, lying at a small distance from each other, consist of one central spot, and several concentric circles, which are more or less distinct, and more or fewer in number, as the metal upon which they are marked is more easy or difficult of fusion, and as a greater or less force is employed. They are composed of dots or cavities, which indicate a real fusion. If the explosion of a battery, issuing from a pointed body, be repeatedly taken on the plain surface of a piece of metal near the point, or be received from the surface on a point, the metal will be marked with a spot, consisting of all the prismatic colours disposed in circles, and formed by scales of the metal separated by the force of the explosion.

Circular Velocity, a term in astronomy signifying | that velocity of a planet, or revolving body, which is measured by the arch of a circle.

Circulating Decimals, called also recurring or repeating decimals, are those in which a sigure or several figures are continually repeated. They are distinguished into single and multiple, and these again into pure and mixed.

A pure single circulate, is that in which one figure only is repeated; as .222 &c, and is marked thus .2.

A pure multiple circulate, is that in which several figures are continually repeated; as .232323 &c, marked .2.3.; and .524524 &c, marked .5.24..

A mixed single circulate, is that which consists of a terminate part, and a single repeating figure; as 4.222 &c, or 4.2.. And

A mixed multiple circulate is that which contains a terminate part with several repeating figures; as 45.5.24..

That part of the circulate which repeats, is called the repetend: and the whole repetend, supposed infinitely continued, is equal to a vulgar fraction, whose numerator is the repeating number, or figures, and its denominator the same number of nines: so .2 is = 2/9; and .2.3. is = 23/99; and .5.24. is = 524/999.

It seems it was Dr. Wallis who first distinctly considered, or treated of infinite circulating decimals, as he himself informs us in his Treatise of Infinites. Since his time many other authors have treated on this part of arithmetic; the principal of these however, to whom the art is mostly indebted, are Messrs. Brown, Cunn, Martin, Emerson, Malcolm, Donn, and Henry Clarke, in whose writings the nature and practice of this art may be fully seen, especially in the last mentioned ingenious author.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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