CLEF

, or Cliff, in Music, a mark at the beginning of the lines of a song, which shews the tone or key in which the piece is to begin. Or, it is a letter marked on any line, which explains and gives the name to all the rest.

Anciently, every line had a letter marked for a Clef; but now a letter on one line suffices; since by this all the rest are known; reckoning up and down, in the order of the letters.

It is called the Clef, or key, because that by it are known the names of all the other lines and spaces; and consequently the quantity of every degree, or interval. But because every note in the octave is called a key, though in another sense, this letter marked is called peculiarly the signed Clef; because, being written on any line, it not only signs and marks that one, but it also explains all the rest. By Clef, therefore, for distinction sake, is meant that letter, signed on a line, which explains the rest; and by key, the principal note of a song, in which the melody closes.

There are three of these signed Clefs, c, f, g. The Clef of the highest part in a song, called treble, or alt, is g, set on the second line counting upwards. The Clef of the bass, or the lowest part, is f on the 4th line upwards. For all the other mean parts, the Clef is c, sometimes on one, sometimes on another line. Indeed, some that are really mean parts, are sometimes set with the g clef. It must however be observed, that the ordinary signatures of Clefs bear little resemblance to those letters. Mr. Malcolm thinks it would be well if the letters themselves were used. Kepler takes great pains to shew, that the common signatures are only cor- ruptions of the letters they represent. The figures of these now are as follow:

Character of the treble Clef.

The mean Clef.

The bass Clef.

The Clefs are always taken fifths to one another. So the Clef f being lowest, c is a fifth above it, and g a fifth above c.

When the place of the Clef is changed, which is not frequent in the mean Clef, it is with a design to make the system comprehend as many notes of the song as possible, and so to have the fewer notes above or below it. So that, if there be many lines above the Clef, and few below it, this purpose is answered by placing the Clef in the first or second line: but if there be many notes below the Clef, it is placed lower in the system. In effect, according to the relation of the other notes to the Clef note, the particular system is taken differently in the scale, the Clef line making one in all the variety.

But still, in whatever line of the particular system any Clef is found, it must be understood to belong to the same of the general system, and to be the same individual note or sound in the scale. By this constant relation of Clefs, we learn how to compare the several particular systems of the several parts, and to know how they communicate in the scale, that is, which lines are unison, and which not: for it is not to be supposed, that each part has certain bounds, within which another must never come. Some notes of the treble, for example, may be lower than some of the mean parts, or even of the bass. Therefore to put together into one system all the parts of a composition written separately, the notes of each part must be placed at the same distances above and below the proper Clef, as they stand in the separate system: and because all the notes that are consonant, or heard together, must stand directly over each other, that the notes belonging to each part may be distinctly known, they may be made with such differences as shall not confound, or alter their significations with respect to time, but only shew that they belong to this or that part. Thus we shall see how the parts change and pass through one another; and which, in every note, is highest, lowest, or unison.

It must here be observed, that for the performance of any single piece, the Clef only serves for explaining the intervals in the lines and spaces: so that it need not be regarded what part of any greater system it is; but the first note may be taken as high or low as we please. For as the proper use of the scale is not to limit the absolute degree of tone; so the proper use of the signed Clef is not to limit the pitch, at which the first note of any part is to be taken; but to determine the tune of the rest, with respect to the first; and considering all the parts together, to determine the relation of their several notes by the relations of their Clefs in the scale: thus, their pitch of tune being determined in a certain note of one part, the other notes of that part are determined by the constant relations of the | letters of the scale, and the notes of the other parts by the relations of their Clefs.

In effect, for performing any single part, the Clef note may be taken in an octave, that is, at any note of the same name; provided we do not go too high, or too low, for finding the rest of the notes of a song. But in a concert of several parts, all the Clefs must be taken, not only in the relations, but also in the places of the system abovementioned; that every part may be comprehended in it.

The natural and artificial note expressed by the same letter, as c and c, are both set on the same line or space. When there is no character of flat or sharp, at the beginning with the Clef, all the notes are natural: and if in any particular place the artificial note be required, it is denoted by the sign of a flat or sharp, set on the line a space before that note.

If a sharp or flat be set at the beginning in any line or space with the Clef, all the notes on that line or space are artificial ones; that is, are to be taken a semitone higher or lower than they would be without such sign. And the same affects all their octaves above and below, though they be not marked so. In the course of the song, if the natural note be sometimes required, it is signified by the character .

COMPASS. Pa. 314, col. 1, after l. 6 from the bottom, add, See also a new one in the Supplement to Cavallo's Treatise on Magnetism.

CONDORCET (John-Anthony-Nicholas de Caritat, Marquis of), member of the Institute of Bologna, of the Academies of Turin, Berlin, Stockholm, Upsal, Philadelphia, Petersbourg, Padua, &c, and secretary of the Paris Academy of Sciences, was born at Ribemont in Picardie, the 17th of September 1743. His early attachment to the sciences, and progress in them, soon rendered him a conspicuous character in the commonwealth of letters. He was received as a member of the Academy of Sciences at 25 years of age, namely, in March 1769, as Adjunct-Mecanician; afterwards, he became Associate in 1710, AdjunctSecretary in 1773, and sole Secretary soon after, which he enjoyed till his death, or till the dissolution of the Academy by the Convention.

Condorcet soon became an author, and that in the most sublime branches of science. He published his Essais d'Analyse in several parts; the first part in 1765 (at 22 years of age); the second, in 1767; and the third, in 1768. These works are chiefly on the Integral Calculus, or the finding of Fluents, and make one volume in 4to.

He published the Eloges of the Academicians or members of the Academy of Sciences, from the year 1666 till 1700, in several volumes. He wrote also similar Eloges of the Academicians who died during the time that he discharged the important office of Secretary to the Academy; as well as the very useful histories of the different branches of science commonly prefixed to the volumes of Memoirs, till the volume for the year 1783, when it is to be lamented that so useful a part of the plan of the Academy was discontinued.

His other memoirs contained in the volumes of the Academy, are the following.

1. Tract on the Integral Calculus; 1765.

2. On the problem of Three Bodies; 1767.

3. Observations on the Integral Calculus; 1767.

4. On the Nature of Infinite Series; on the Extent of the Solutions which they give; and on a new method of Approximation for Differential Equations of all Orders; 1769.

5. On Equations for Finite Differences; 1770.

6. On Equations for Partial Differences; 1770.

7. On Differential Equations; 1770.

8. Additions to the foregoing Tracts; 1770.

9. On the Determination of Arbitrary Functions which enter the Integrals of Equations to Partial Differences; 1771.

10. Reflexions on the Methods of Approximation hitherto known for Differential Equations; 1771.

11. Theorem concerning Quadratures; 1771.

12. Inquiry concerning the Integral Calculus; 1772.

13. On the Calculation of Probabilities, part 1 and 2; 1781.

14. Continuation of the same, part 3; 1782.

15. Ditto, part 4; 1783.

16. Ditto, part 5; 1784.

Condorcet had the character of being a very worthy honest man, and a respectable author, though perhaps not a first-rate one, and produced an excellent set of Eloges of the deceased Academicians, during the time of his secretaryship. A late French political writer has observed of him, that he laboured to succeed to the literary throne of d'Alembert, but that he cannot be ranked among illustrious authors; that his works have neither animation nor depth, and that his style is dull and dry; that some bold attacks on religion and declamations against despotism have chiefly given a degree of fame to his writings.

On the breaking out of the troubles in France, Condorcet took a decided part on the side of the people, and steadily maintained the cause he had espoused amid all the shocks and intrigues of contending parties; till, under the tyranny of Robespierre, he was driven from the convention, being one of those members proscribed on the 31st of May 1793, and he died about April 1794. The manner of his death is thus described by the public prints of that time. He was obliged to conceal himself with the greatest care for the purpose of avoiding the fate of Brissot and the other deputies who where executed. He did not, however, attempt to quit Paris, but concealed himself in the house of a female, who, though she knew him only by name, did not hesitate to risk her own life for the purpose of preserving that of Condorcet. In her house he remained till the month of April 1794, when it was rumoured that a domiciliary visit was to be made, which obliged him to leave Paris. Although he had neither passport nor civic card, he escaped through the Barrier, and arrived at the Plain of Mont rouge, where he expected to find an asylum in the country-house of an intimate friend. Unfortunately this friend had set out for Paris, where he was to remain for three days.—During all this period, Condorcet wandered about the fields and in the woods, | not daring to enter an inn on account of not having a civic card. Half dead with hunger, fatigue, and fear, and scarcely able to walk on account of a wound in his foot, he passed the night under a tree.

At length his friend returned, and received him with great cordiality; but as it was deemed imprudent that he should enter the house in the day-time, he returned to the woods till night. In this short interval between morning and night his caution forsook him, and he resolved to go to an inn for the purpose of procuring food. He went to an inn at Clamars, and ordered an omlette. His torn clothes, his dirty cap, his meagre and pale countenance, and the greediness with which he devoured the omlette, fixed the attention of the persons in the inn, among whom was a member of the Revolutionary Committee of Clamars. This man conceiving him to be Condorcet, who had effected his escape from the Bicetre, asked him whence he came, whither he was going, and whether he had a passport? The confused manner in which he replied to these questions, induced the member to order him to be conveyed before the Committee, who, after an examination, sent him to the district of Boury la Reine. He was there interrogated again, and the unsatisfactory answers which he gave, determined the directors of the district to send him to prison on the succeeding day.—During the night he was confined in a kind of dungeon. On the next morning, when his keeper entered with some bread and water for him, he found him stretched on the ground without any signs of life.

On inspecting the body, the immediate cause of his death could not be discovered, but it was conjectured that he had poisoned himself. Condorcet indeed always carried a dose of poison in his pocket, and he said to the friend who was to have received him into his house, that he had been often tempted to make use of it, but that the idea of a wife and daughter, whom he loved tenderly, restrained him. During the time that he was concealed at Paris, he wrote a history of the Progress of the Human Mind, in two volumes.

CUBICS. The method of resolving all the cases of Cubic equations by the tables of sines, tangents and secants, are thus given by Dr. Maskelyne, p. 57, Taylor's Logarithms.

“The following method is adapted to a Cubic equation, wanting the second term; therefore, if the equation has the second term, it must be first taken away in the usual manner. There are four forms of Cubic equations wanting the second term, whose roots, according to known rules equivalent to Cardan's, are as follow:

The roots of the first and second forms are negatives of each other; and those of the third and fourth are also negatives of each other. The first and second forms have only one root each. The third and fourth forms have also only one root each, when the quadratic surd √(q²/4 - (p³/27)) is possible; but have three roots each when that surd is impossible.

The roots of all the four forms may, in all cases, be easily computed as follows:

Forms 1st and 2d. Put ; where the upper sign belongs to the first form, and the lower sign to the second form.

Forms 3d and 4th. Put 2/q X ―(p/3))^3/2 if less than unity, else its reciprocal . Then,

Case 1st. 2/q X ―(p/3))^3/2 < unity. Put ; where the upper sign belongs to the third form, and the lower sign to the fourth form.

Case 2d. 2/q X ―(p/3))^3/2 > unity. Then x has three values in each form, viz, ; where the upper signs belong to the third form, and the lower signs to the fourth form. |

By Logarithms.

Eorms 1st and 2d. ; and x will be affirmative in the first form, and negative in the second form.

Forms 3d and 4th. (3/2) X log.p/3 + 10 - log.q/2 being less than 10 (which is case first) or log.q/2 + 10 - (3/2) X log.p/3 being less than 10 (which is case 2d) = log. cos. z.

Case 1st. ; and x will be affirmative in the third form, and negative in the 4th form.

Case 2d. Here x has three values. ; where the upper signs belong to the third form, and the lower signs to the fourth form; that is, the first value of x in the third form is positive, and its second and third values negative; and the first value of x in the fourth form is negative, and its second and third values affirmative.”