# DEGREE

, in Algebra, is used in speaking of equations, when they are said to be of such a degree according to the highest power of the unknown quantity. If the index of that power be 2, the equation is of the 2d degree; if 3, it is of the 3d degree, and so on.

Degree

, in Geometry or Trigonometry, is the 360th part of the circumference of any circle; for every circle is considered as divided into 360 parts, called degrees; which are marked by a small ° near the top of the figure; thus 45° is 45 degrees.

The degree is subdivided into 60 smaller parts, called minutes, meaning first minutes; the minute into 60 others, called seconds; the second into 60 thirds; &c. Thus 45° 12′ 20″ are 45 degrees, 12 minutes, 20 seconds.

The magnitude or quantity of angles is accounted in degrees; for because of the uniform curvature of a circle in all its parts, equal angles at the centre are subtended by equal arcs, and by similar arcs in peripheries of different diameters; and an angle is said to be of so many degrees, as are contained in the arc of any circle comprehended between the legs of the angle, and having the angular point for its centre. Thus we say an angle of 90°, or of 45° 24′, or of 12° 20 30″. It is also usual to say, such a star is mounted so many degrees above the horizon, or declines so many degrees from the equator; or such a town is situate in so many degrees of latitude or longitude.—A sign of the ecliptic, or zodiac, contains 30 degrees.

The division of the circle into 360 degrees is usually ascribed to the Egyptians, probably from the circle of the sun's annual course, or according to their number of days in the year, allotting a degree to each day. It is a convenient number too, as admitting of a great many aliquot parts, as 2, 3, 4, 5, 6, 8, 9, &c. The sexagesimal subdivision, however, has often been condemned as improper, by many eminent mathematicians, as Stevinus, Oughtred, Wallis, Briggs, Gellibrand, Newton, &c; who advise a decimal division instead of it, or else that of centesms; as the degree into 100 parts, and each of these into 100 parts again, and so on. Stevinus even holds, that this division of the circle which he contends for, obtained in the wise age, in sæculo sapienti. Stev. Cosmog. lib. 1, def. 6. And several large tables of sines &c have been constructed according to that plan, and published, by Briggs, Newton, and others. And I myself have carried the idea still much farther, in a memoir published in the Philos. Trans. of 1783, containing a proposal for a new division of the quadrant, viz, into equal decimal parts of the radius; by which means the degrees or divisions of the arch would be the real lengths of the arcs, in terms of the radius: and I have since computed those lengths of the arcs, with their sines, &c, to a great extent and accuracy.

Degree of Latitude, is the space or distance on the meridian through which an observer must move, to vary his latitude by one degree, or to increase or diminish the distance of a star from the zenith by one degree; and which, on the supposition of the perfect sphericity of the earth, is the 360th part of the meridian.

The quantity of a degree of a meridian, or other great circle, on the surface of the earth, is variously determined by different observers: and the methods made use of are also various.

Eratosthenes, 250 years before Christ, first determined the magnitude of a degree of the meridian, between Alexandria and Syene on the borders of Ethiopia, by measuring the distance between those places, and comparing it with the difference of a star's zenith distances at those places; and found it 694 4/9 stadia. |

Posidonius, in the time of Pompey the Great, by means of the different altitudes of a star near the horizon, taken at different places under the same meridian, compared in like manner with the distance between those places, determined the length of a degree only 600 stadia.

Ptolomy fixes the degree at 68 2/3 Arabic miles, counting 7 1/2 stadia to a mile. The Arabs themselves, who made a computation of the diameter of the earth, by measuring the distance of two places under the same meridian, in the plains of Sennar, by order of Almamon, make it only 56 miles. Kepler, determining the diameter of the earth by the distance of two mountains, makes a degree 13 German miles; but his method is far from being accurate. Snell, seeking the diameter of the earth from the distance between two parallels of the equator, finds the quantity of a degree, by one method 57064 Paris toises, or 342384 feet; by another meth. 57057 toises, or 342342 feet. The mean between which two numbers, M. Picard found by mensuration, in 1669, from Amiens to Malvoisin, the most certain, and he makes the quantity of a degree 57060 toises, or 342360 feet. However, M. Cassini, at the king's command, in the year 1700, repeated the same labour, and measuring the space of 6° 18′, from the observatory at Paris, along the meridian, to the city of Collioure in Roussillon, that the greatness of the interval might diminish the error, found the length of the degree equal to 57292 toises, or 343742 Paris feet, amounting to 365184 English feet.

And with this account nearly agrees that of our countryman Norwood, who, about the year 1635, measured the distance between London and York, and found that distance 905751 English feet; the difference of latitude being 2° 28′, hence, he determined the quantity of one degree at 367196 English feet, or 57300 Paris toises, or 69 miles, 288 yards. See Newt. Princ. Phil. prop. 19; and Hist. Acad. Scienc. anno 1700, pa. 153.

M. Cassini, the son, completed the work of measuring the whole arc of the meridian through France, in 1718. For this purpose he divided the meridian of France into two arcs, which he measured separately. The one from Paris to Collioure gave him 57097 toises; the other from Paris to Dunkirk 56960; and the whole are from Dunkirk to Collioure 57060, the same as M. Picard's.

M. Muschenbroek, in 1700, resolving to correct the errors of Snell, found by particular observations, that the degree between Alcmaer and Bergen-op-zoom contained 57033 toises.

Messieurs Maupertuis, Clairaut, Camus, Monnier, and Outheir of France, were sent on a northern expedition, and began their operations, assisted by M. Celsus, an eminent astronomer of Sweden, in Swedish Lapland, in July 1736, and sinished them by the end of May following. They obtained the measure of that degree, whose middle point was in lat. 66° 20′ north, and found it 57439 toises, when reduced to the level of the sea. About the same time another company of philosophers was sent to South America, viz, Messieurs Godin, Bouguer, and Condamine of France, to whom were joined Don Jorge Juan, and Don An- tonio de Ulloa of Spain. They left Europe in 1735, and began their operations in the province of Quito in Peru, about October 1736, and finished them, after many interruptions, about 8 years after. The Spanish gentlemen published a separate account, and assign for the measure of a degree of the meridian at the equator 56768 toises. M. Bouguer makes it 56753 toises, when reduced to the level of the sea; and M. Condamine states it at 56749 toises.

M. Caille, being at the Cape of Good Hope in 1752, found the length of a degree of the meridian in lat. 33° 18′ 30″ south, to be 57037 toises. In 1755, father Boscovich found the length of a degree in lat. 43° north to be 56972 toises, as measured between Rome and Rimini in Italy. In the year 1740, Messrs Cassini and La Caille again examined the former measures in France, and, after making all the necessary corrections, found the measure of a degree, whose middle point is in lat. 49° 22′ north, to be 57074 toises; and in the lat. of 45°, it was 57050 toises.

In 1764, F. Beccaria completed the measurement of a portion of the meridian near Turin; from which it is deduced that the length of a degree, whose middle lat. is 44° 44′ north, is 57024 Paris toises.

At Vienna, 3 degrees of the meridian were measured; and the medium, for the latitude of 47° 40′ north may be taken at 57091 Paris toises. See an account of this measurement, by father Joseph Liesganig, in the Philos. Trans. 1768, pa. 15.

Finally, in the same vol. too is an account of the measurement of a part of the meridian in Maryland and Pensilvania, North America, 1766, by Messrs Mason and Dixon; from which it follows that the length of a degree whose middle point is 39° 12′ north, was 363763 English feet, or 56904 1/2 Paris toises.

Hence, from the whole we may collect the following table of the principal measures of a degree in different parts of the earth, as measured by different persons, viz,

 Mean Latitude. Length of a Degree in Paris toises. Names of the Measurers. Years of Measurement. 66° 20′ N 57422 Maupertuis &c 1736 & 1737 49 23 N 57074 { Maupertuis &c and Cassini 1739 & 1740 47 40 N 57091 Liesganig 1766 45 0 N 57028 Cassini 1739 & 1740 44 44 N 57069 Beccaria 1760 to 1764 43 0 N 56979 { Boscovich and Le Maire 1752 39 12 N 56888 Mason & Dixon 1764 to 1768 0 0 56750 { Bouguer and Condamine 1736 to 1744 33 18 S 57037 La Caille 1752

The method of obtaining the length of a degree of the terrestrial meridian, is to measure a certain distance upon it by a series of triangles, whose angles may be found by actual observation, connected with a base, whose length may be taken by an actual survey, or otherwise; and then to observe the different altitudes | of some star at the two extremities of that distance, which gives the difference of latitude between them: then, by proportion, as this difference of latitude is to one degree, so is the measured length to the length of one degree of the meridian sought. This method was first practised by Eratosthenes, in Egypt. See GGEOGRAPHY, and the beginning of this article.

Degree of longitude, is the space between two meridians that make an angle of 1° with each other at the poles; the quantity or length of which is variable, according to the latitude, being every where as the cosine of the latitude; viz, as the cosine of one lat. is to the cosine of another, so is the length of a degree in the former lat. to that in the latter; and from this theorem is computed the following Table of the length of a degree of long. indifferent latitudes, supposing the earth to be a globe.

 Degr. lat. English miles. Degr. lat. English miles. Degr. lat. English miles. 0 69.07 31 59.13 61 33.45 1 69.06 32 58.51 62 32.40 2 69.03 33 57.87 63 31.33 3 68.97 34 57.20 64 30.24 4 68.90 35 56.51 65 29.15 5 68.81 36 55.81 66 28.06 6 68.62 37 55.10 67 26.96 7 68.48 38 54.37 68 25.85 8 68.31 39 53.62 69 24.73 9 68.15 40 52.85 70 23.60 10 67.95 41 52.07 71 22.47 11 67.73 42 51.27 72 21.32 12 67.48 43 50.46 73 20.17 13 67.21 44 49.63 74 19.02 14 66.95 45 48.78 75 17.86 15 66.65 46 47.93 76 16.70 16 66.31 47 47.06 77 15.52 17 65.98 48 46.16 78 14.35 18 65.62 49 45.26 79 13.17 19 65.24 50 44.35 80 11.98 20 64.84 51 43.42 81 10.79 21 64.42 52 42.48 82 9.59 22 63.97 53 41.53 83 8.41 23 63.51 54 40.56 84 7.21 24 63.03 55 39.58 85 6.00 25 62.53 56 38.58 86 4.81 26 62.02 57 37.58 87 3.61 27 61.48 58 36.57 88 2.41 28 60.93 59 35.54 89 1.21 29 60.35 60 34.50 90 0.00 30 59.75

Note, This table is computed on the supposition that the length of the degrees of the equator are equal to those of the meridian at the medium latitude of 45°, which length is 69 1/15 English miles.

The expressions Latitude and Longitude, are borrowed from the ancients, who happened to be acquainted with a much larger extent of the earth in the direction east and west, than in that of north and south; the former of which therefore passed, with them, for the length of the earth, or longitude, and the latter for the breadth or shorter dimension, viz, the latitude.

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

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