tion.

Gombina to be 62,044,840,173,323,943,936,000. Now the inches in a fquare yard being 1296, that number multiplied by roo gives 129,600, which is the number of letters each fquare yard will contain; therefore if we divide 62,044,840,173,323,943,936,coo by 129600 the quotient, which is 478,741,050,720,092,160, will be the number of yards required, to contain the above mentioned number of permutations. But as all the 24 letters are contained in every permutation, it will require a space 24 times as large; that is, 11,489,785,217,282,211,840. Now the number of fquare yards contained on the furface of the whole earth is but 617,197,435,008,000, therefore it would require a furface 18620 times as large as that of the earth to write all the purmutations of the 24 letters in the fize above mentioned.

III. To find how many different ways the eldest hand at piquet may take in his five cards. The eldeft hand having 12 cards dealt him, there remain 20 cards, any five of which may be in those he takes in; confequently we are here to find how many ways five cards may be taken out of 20. Therefore, by aphorifm I. if we multiply 20, 19, 18, 17, 16, into each other, which will make 1860480, and that number be divided by 1, 2, 3, 4, 5, multiplied into each other, which make 120, the quotient, which is 15504, will be the number of ways five cards may be taken out of 20. From hence it follows, that it is 15503 to 1, that the eldest hand does not take in any five certain cards.

IV. To find the number of deals a perfon may play at the game of whift, without ever holding the fame cards twice. The number of cards played with at

whilst being 52, and the number dealt to each perfon being 13, it follows, that by taking the fame method as in the last experiment, that is, by multiplying 52 by 51, 50, &c. fo on to 41, which will make 3,954,242,643,911,239,680,000, and then dividing that fum by 1, 2, 3, &c. to 13, which will make 6,227,020,800, the quotient, which is635,013,559,600 will be the number of different ways 13 cards may be taken out of 52, and confequently the number fought.

i Rank

line A a confifts of the first 12 numbers. The line The construction of this table is very fimple. The Ab confifts every where of units; and fecond term 3, in the preceding rank: the third term 6, in that line, of the line B c, is compofed of the two terms 1 and 2 is formed of the two terms 3 and 3 in the preceding rank: and fo of the reft; every term, after the first, being compofed of the two next terms in the preceding rank and by the fame method it may be continued to any number of ranks. To find by this table how often any number of things can be combined in another number, under 13, as fuppofe 5 cards out of 56, and that is the number required. 8; in the eighth rank look for the fifth term, which is

Though we have fhown in the foregoing problems whatever, yet as this table anfwers the fame purpose, the manner of finding the combination of all numbers for fmall numbers, by infpection only, it will be found lowing examples. useful on many occafions; as will appear by the foldifferent founds

V. To find how

1. The

many duced by ftriking on a harpsichord two or more of be may prothe feven natural notes at the fame time. combinations of two in feven, by the foregoing triangle are

2. The combinations of in
3 7, are
3. The combinations of 4 in 7, are
4. The combinations of 5, are
5. The combinations of 6, are

6. The feven notes all together once

Therefore the number of all the founds will be

21

35

35

21

7

120

fame dimenfion, and divide them diagonally, that is VI. Take four fquare pieces of pafteboard, of the by drawing a line from two oppofite angles, as in the figures, into 8 triangles; paint 7 of these triangles with the primitive colours, red, orange, yellow, green, blue, indigo, violet, and let the eighth be white. To find how many chequers or regular four-fided figures, different either in form or colour, may be made out of thofe eight triangles. Firft, by combining two of thefe triangles, there may be formed either the triangular fquare A, or the inclined fquare B called a rhomb. Secondly, by combining four of the triangles, the large fquare C may be formed; or the long fquare D, called a parallelogram.

7. 21. 35.

35. 21.

7. I

8.

a 12. 66. 220. 495. 792. 924. 792. 495. 220. 66. 12. 1 ḥ

11. 55. 165. 330. 462. 462. 330. 165. 55. 11. 1

10. 45. 120. 210. 252. 210. 120. 45. 10. I

9. 36. 84. 126. 126. 84. 36. 9. 1

28. 56. 70. 56. 28. 8. 1

It is a matter of indifference what numbers are made Combination. use of in forming these tables. We fhall here confine ourselves to fuch as are applicable to the subsequent experiments. Any one may conftruct them in fuch manner as is agreeable to the purposes he intends they shall answer.

VII. A man has 12 different forts of flowers, and a large number of each fort. He is defirous of fetting them in beds or flourishes in his parterre: Six flowers in fome, 7 in others, and 8 in others; fo as to have the greateft variety poffible; the flowers in no two beds to be the fame. To find how many beds he muft have. 1. The combinations of 6 in 12 by the laft rank of the triangle, are

2. The combinations of 7 in 12, are 3. The combinations of 8 in 12, are Therefore the number of beds must be

924

792

495

2211

VIII. To find the number of chances that may be thrown on two dice. As each die has 6 faces, and as every face of one die may be combined with all the faces of the other, it follows, that 6 multiplied by 6, that is 36, will be the number of all the chances; as is alfo evident from the following table:

It appears by this table, 1. That the number of chances for each point continually increases to the point of feven, and then continually decreafes till 12: therefore if two points are propofed to be thrown, the equality, or the advantage of one over the other, is clearly vifible (a). 2. The whole number of chances on the dice being 252, if that number be divided by 36, the number of different throws on the dice, the quotient is 7: it follows therefore, that at every throw there is an equal chance of bringing feven points. 3. As there are 36 chances on the dice, and only 6 of them doublets, it is 5 to 1, at any one throw, against throwing a doublet.

By the fame method the number of chances upon any number of dice may be found: for if 36 be multiplied by 6, that product, which is 216, will be the chances on 3 dice; and if that number be multiplied by 6, the product will be the chances on 4 dice, &c.

COMBINATIONS of the Cards. The following experiments, founded on the doctrine of combinations, may poffibly amufe a number of our readers. The tables given are the bafis of many experiments, as well on numbers, letters, and other subjects, as on the cards; but the effect produced by them with the laft is the moft furprising, as that which fhould feem to prevent any collufion, that is the shuffling of the cards, is on the contrary the cause from whence it proceeds.

To make them, for example, correfpond to the nine digits and a cipher, there must be ten cards, and at the top of nine of them must be written one of the digits, and on the tenth a cipher. Thefe cards muft be placed upon each other in the regular order, the number 1 being on the firft, and the cipher at bottom, You then take the cards in your left hand, as is commonly done in fhuffling, and taking off the two top cards, I and 2, you place the two following, 3 and 4, upon them ; and under thofe four cards the three following 5, 6, and 7: at the top you put the cards 8 and 9, and at the bottom the card marked o; conftantly placing in fucceffion 2 at top and 3 at bottom: And they will then be in the following order:

5.6.3.4.7.8.9.1.2.0 1.2.3.4.5.6.7.8.9.0

6

7

the cards return to the order in which they were first It is a remarkable property of this number, that placed, after a number of fhuffles, which added to the number of columns that never change the order, is fhuffles is 7, and the number of columns in which the equal to the number of cards. Thus the number of cards marked 3, 4, &c. never change their places is 3, which are equal to 10, the number of the cards. This property is not common to all numbers; the cards and fometimes in a greater number of fhuffles than that fometimes returning to the first order in a lefs number,

of the cards.

Colymbus. have thought right to figure it, as a fpecies, if not new, at least as not generally known; and probably, from the circumftance of its fituation in the painting, may prove one of the birds used on this occafion.

Com.

of the halcyon's neft may, in fome measure, be vindi- Cufymbne, cated. It is a careful nurfe of its young; being obferved to feed them moft affiduoufly, commonly with fmall eels; and when the infant brood are tired, the parent will carry them either on its back or under its wings. It preys on fith, and is almost perpetually diving; it does not fhow much more than the head above water: and is very difficult to be fhot, as it darts down on the leaft appearance of danger. It is never feen on land; and, though difturbed ever fo often, will not fly farther than the end of the lake. Its fkin is out of feafon about February, lofing then its bright colour; and in the breeding time its breaft is almoft bare. The flesh is exceffively rank.

8. The ftellatus, or fpeckled diver, a fpecies lefs than the former, weighs two pounds and a half: and is 27 inches in length and three feet nine in breadth. The bill is three inches long, bending a trifle upwards; and is of a pale horn-colour, the top of the upper maudible dafky; the head is dufky, dotted with grey; hind part of the neck plain dufky; the fides under the eye, the chin, and throat, white; fore part of the neck very pale afh-colour; back dusky, marked with oval fpots of white; fides of the breaft and body the fame, but fmaller; the spots on the rump and tail minute; breast and under parts white; quills dufky; legs brown; webs and claws palè. This bird is pretty frequent in England; fufficiently fo on the river Thames, where it is called by the fishermen fprat loon, being often feen in vaft numbers among the fhoals of that fifh, diving after them, and frequently approaching very near the boats while fishing. It is common about the Baltic and the White Sea, but not obferved in other parts of Ruffia, yet is a native of Kamtschatka. It lays two eggs, in the grafs, on the borders of lakes not far from the fea; they are exactly oval, the fize of thofe of a goose, dusky, marked with a few black spots. Thefe are alfo frequent about the fish ponds in France, except they are frozen, when they betake themselves to the rivers. This and the two laft vifit New York in winter, but return very far north to breed.

9. The cryftatus, crefted diver, or cargoofe, weighs two pounds and an half. Its length is 21 inches, the breadth 30; the bill is two inches and a quarter long, red at the bafe, and black at the point; between the bill and the eyes is a stripe of black naked fkin; the irides are of a fine pale red; the tongue is a third part fhorter than the bill, slender, hard at the end, and a little divided; on the head is a large dufky creft, feparated in the middle. The cheeks and throat are furrounded with a long pendent ruff, of a bright tawney colour, edged with black; the chin is white; from the bill to the eye is a black line, and above that a white one; the hind part of the neck and the back are of a footy hue; the rump, for it wants a tail, is covered with long foft down. The covert-feathers on the fecond and third joints of the wing, and the under coverts, are white; all the other wing-feathers, except the fecondaries, are dufky, thofe being white; the breaft and belly are of a most beautiful filvery white, gloffy as fattin': the outfide of the legs and the bottom of the feet are dufky; the infide of the legs and the toes of a pale green. Thefe birds frequent the meres of Shropfhire and Cheshire, where they breed; and the great fen of Lincolnshire, where they are called gaunts. Their fins are made into tippets, and fold at as high a price as those which come from Geneva. This fpecies lays four egge of a white colour, and the fame fize with thofe of a pigeon. The neft is formed of the roots of bugbane, ftalks of water-lily, pond-weed, and water-violet, floating independent among the reeds and flags; the water penetrates it, and the bird fits and hatches the eggs in that wet condition; the neft is fometimes blown from among the flags into the middle of the water: in thefe circumstances the fable VOL. V. Part I.

10. The urinator, or tippet-grebe, thought by Mr Latham not to be a different fpecies from the former, being only fomewhat lefs, and wanting the creft and ruff. The fides of the neck are ftriped downwards from the head with narrow lines of black and white : in other refpects the colours and marks agree with that bird. This fpecies has been shot on Rostern Mere in Cheshire. It is rather fcarce in England, but is common in the winter time on the lake of Geneva. They appear there in flocks of 10 or 12; and are killed for the fake of their beautiful skins. The under fide of them being dreffed with the feathers on, are made into muffs and tippets: each bird fells for about 14 fhillings.

11. The auritus, eared grebe, or dob-chick, isin length one foot to the rump; the extent is 22 inches; the bill black, flender, and flightly recurvated; the irides crimson; the head and neck are black; the throat fpotted with white; the whole upper fide of a blackish brown, except the ridge of the wing about the first joint, and the fecondary feathers, which are white; the breaft, belly, and inner coverts of the wings are white; the fubaxillary feathers, and fome on the fide of the rump, ferruginous. Behind the eyes, on each fide, is a tuft of long, loofe, ruft coloured feathers hanging backwards; the legs are of a dusky green. They inhabit the fens near Spalding where they breed. No external difference is to be observed between the male and the female of this fpecies. They make their neft not unlike that of the former; and lay four or five small eggs.

12. The horned grebe, is about the fize of a teal; weight, one pound; length, one foot; breadth, 16 inches. Bill one inch, duky; head very full of feathers, and of a gloffy deep green, nearly black: thro' each eye is a ftreak of yellow feathers, elongated into a tuft as it paffes to the hind head: the upper part of the neck and back is a dufky brown; the fore part of the neck and breaft, dark orange red: the leffer wing coverts, cinerous; the greater and quills, black; middle ones, white belly, gloffy white; legs, cinerous blue before, pale behind. It inhabites Hudfon's bay; and firft appears in May, about the fresh waters. lays from two to four white eggs in June, among the aquatic plants; and is faid to cover theem when abroad, It retires fouth in autumn; appears then at New York, ftaying till fpring, when it returns to the north. For its vaft quickness in diving, it is called the water-witch. At Hudfon's bay, it is known by the name of seekeep. See Plate CXLIII.

It

COM, a town of Afia in the empire of Perfia, and province of Iracagemi. It is a large populous place,

but

Coma but has fuffered greatly by the civil wars. E. Long. Combina- 49.1. N. Lat. 34. 0.

tion.

COMA, or COMA-VIGIL, a preternatural propenfity to fleep, when, nevertheless, the patient does not fleep, or if he does, awakes immediately without any relief. See MEDICINE-Index.

COMA Berenices, Berenice's hair, in aftronomy, a modern conftellation of the northern hemifphere, compofed of unformed ftars between the Lion's tail and Bootes. This conftellation is faid to have been formed by Conon, an aftronomer, in order to confole the queen of Ptolemy Evergetes for the lofs of a lock of her hair, which was ftolen out of the temple of Venus, where he had dedicated it on account of a victory obtained by her husband. The stars of this conftellation, in Tycho's Catalogue, are fourteen; in Hevelius's, twenty-one; and in the Britannic Catalogue, forty-three. COMA Somnolentum, is when the patient continues in a profound fleep; and, when awakened, immediately relapfes, without being able to keep open his eyes. COMARUM, MARSH-CINQUEFOIL: A genus of the polygynia order, belonging to the icofandria clafs of plants; and in the natural method ranking under the 35th order, Senticofa. The calyx is decemfid; the petals five, lefs than the calyx; the receptacle of the feeds ovate, spongy, and perfifting. There is but one species, a native of Britain. It rifes about two feet high, and bears fruit fomewhat like that of the ftrawberry. It grows naturally in bogs, fo is not easily preferved in gardens. The root dyes a dirty red. The Irish rub their milking pails with it, and it makes the milk appear thicker and richer. Goats eat the herb; cows and sheep are not fond of it; horfes and swine refuse it.

COMB, an inftrument to clean, untangle, and drefs flax, wool, hair, &c.

Combs for wool are prohibited to be imported into England.

COMB is alfo the creft, or red fleshy tuft, growing upon a cock's head.

COMBAT, in a general sense, denotes an engagement, or a difference decided by arms. See BATTLE. COMBAT, in our ancient law, was a formal trial of fome doubtful caufe or quarrel, by the fwords or baftons of two champions. This form of proceeding was very frequent, not only in criminal but in civil caufes; being built on a fuppofition that God would never grant the victory but to him who had the beft right. The laft trial of this kind in England was between Donald lord Ray appellant, and David Ramfay, Efq; defendant, when, after many formalities, the matter was referred to the King's pleafure. See the article BATTLE.

COMBER, or CUMBER (Thomas), an eminent divine born at Weftram in Kent, in 1645, was educated at Cambridge; created doctor of divinity; and, after feveral preferments in the church, was made dean of Durham. He was chaplain to Anne princefs of Denmark, and to king William and queen Mary. He was author of feveral works, viz. 1. A fcholaftical hiftory of the primitive and general ufe of Liturgies. 2. A Companion to the Altar. 3. A brief difcourfe upon the offices of baptifm, catechifm, and confirmation. He died in 1699, aged 55.

COMBINATION, properly denotes an affemblage of feveral things, two by two.

COMBINATION, in mathematics, is the variation or Combina alteration of any number of quantities, letters, or the like, in all the different manners poffible. See CHANGES.

Aphorifms. I. In all combinations, if from an arithmetic decreafing feries, whose first term is the number out of which the combinations are to be formed, and whose common difference is 1, there be taken as many terms as there are quantities to be combined, and thefe terms be multiplied into each other; and if from the feries 1, 2, 3, 4, &c. there may be taken the fame number of terms, and they be multiplied into each other, and the first product be divided by the fecond; the quotient will be the number of combinations required. Therefore, if you would know how many ways four quantities can be combined in seven, multiply the first four terms of the feries, 7, 6, 5, 4, &c. together, and divide the product, which will be 840, by the product of the first four terms of the feries, 1, 2, 3, 4, &c. which is 24, and the quotient 35 will be the combinations of 4 in 7. II. In all permutations, if the feries 1, 2, 3, 4, &c. be continued to as many terms as there are quantities to be changed, and thofe terms be multiplied into each other; the product will be the number of permutations fought. Thus, if you would know how many permutations can be formed with five quantities, multiply the terms 1, 2, 3, 4, 5, together, and the product 120 will be the number of all the permutations.

Problems. I. To find the number of changes that may be rung on 12 bells. It appears by the fecond aphorifin, that nothing is more neceffary here than to multiply the numbers from 1 to 12 continually into each other, in the following manner, and the laft product will be the number fought.

tion.

to

Combina for 32 numbers, and dispose these 32 cards in the foltion. lowing order, by that column.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 OIERG CANTP INTAI S 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 T MEHS DIN NOYNTEIS The cards being thus difpofed, fhuffle them once, and deal them two and two; when one of the parties will neceffarily have the queftion, and the other the answer.

Instead of letters you may write words upon the 32 cards, 16 of which may contain a queftion, and the remainder the anfwer; or what other matter you pleafe. If there be found difficulty in accommodating the words to the number of cards, there may be two or more letters or fyllables written upon one card.

V. "The five beatitudes." The five bleffings we will fuppofe to be, 1. Science, 2. Courage, 3. Health, 4. Riches, and 5. Virtue. Thefe are to be found upon cards that you deal, one by one, to five perfons. Firft write the letters of these words fucceffively, in the order they ftand, and then add the numbers here annexed to them.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 LH NATER EUA CRG TIU 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 EE CIICHSOHREEV SC Next, take a pack of cards, and write on the four first the word Science; on the four next the word Courage; and fo of the reft.

Matters being thus prepared, you show that the cards on which the letters are written convey no meaning. Then take the pack on which the words are written, and spreading open the firft four cards, with their backs upward, you defire the firft perfon to choose one. Then clofe thofe cards and fpread the next four to the second perfon; and fo to all the five; telling them to hold up their cards left you fhould have a confederate in the room.

You then fhuffle the cards, and deal them one by one, in the common order, beginning with the perfon who chose the first card, and each one will find in his hand the fame word as is written on his card. You will obferve, that after the fixth round of dealing, there will be two cards left, which you give to the firft and fecond perfons, as their words contain a letter more than the others.

VI. "The cards of the game of piquet being mixed together, after fhuffling them, to bring, by cutting them, all the cards of each fuit together." The order in which the cards must be placed to produce the effect defired being established on the fame principle as that explained in the experiment II. except that the fhuf

12 Queen clubs 13 Eight? hearts 14 Seven wide card

} fpades

17 King clubs hearts

18 Ten 19 Nine

20 Seven clubs

21 Ace diamonds 22 Knave fpades 23 Queen hearts

24 Knave hearts 25 Ace spades 26 King diamonds

27

Nine clubs

28 Ace

29 King S

hearts

30 Eight clube

15 Ten 31 King 16 Nine 32 Queen Spades You then fhuffle the cards, and cutting at the wide card, which will be the feven of hearts, you lay the eight cards that are cut, which will be the fuit of hearts, down on the table. Then shuffling the remaining cards a fecond time, you cut at the fecond wide card, which will be the feven of fpades, and lay, in like manner, the eight fpades down on the table. You fhuffle the cards a third time, and offering them to any one to cut, he will naturally cut them at the wide card (D), which is the feven of diamonds, and confequently divide the remaining cards into two equal parts, one of which will be diamonds and the other clubs.

VII. "The cards at piquet being all mixed together, to divide the pack into two unequal parts, and name the number of points contained in each part." You are first to agree that each king, queen, and knave shall count, as ufual, 10, the ace 1, and the other cards according to the number of the points. Then difpofe the cards, by the table for 32 numbers, in the following order, and obferve that the laft card of the first divi fion must be a wide card.

Order of the cards before fhuffling.
17 Nine diamonds
18 Ace fpades

1 Seven hearts

2 Nine clubs 3 Eight hearts 4 Eight

5

Knave 6 Ten

fpades

7 Queen
8 Aceclube

9 Ace hearts
wide card

10 Nine hearts
11 Queen Spades
12 Knave clubs
13 Ten diamonds

19 Ten clubs 20 Knave

21 Eight

22 King

23 Seven Spades

24 Seven

}

diamond

25 Queen diamonds