of this series be written in one line, and the negative ones in another, the arch will be &c. A, B, C, &c. respectively, the series will be expressed in the concise and elegant notation of sir Isaac Newton, viz. 3nxC m+ 3n nz A 2nz B -+ + &c. which is well adapted to arithmetical calculation. "6. I come now to the transformation proposed, which will appear very easy, as soon as the common series, expressing the length of an arch in terms of its tangent, is properly arranged. "If the radius of a circle be 1, and the tangent of an arch of it be called, it is well known that 13 15 17 the length of that arch will be-t--+--3 + + + + &c. 21 29 37 All which series are evidently of the first form in article 2. and therefore their values may be expressed in the second form there given, or more neatly by the Newtonian notation mentioned in article 5. In each of these series the value of n is 8. in the first series, is 1; in the second series, is 5; And the value of m,in the third series, is 3; in the fourth series, is 7. "If now we take = , the tangent of 30°, which was chosen by Dr. Hailey, we shall have the arch of 30° 7. All these new series, it is evident, converge somewhat swifter than by the powers of 80. For in the first series, which has the slowest convergency, the coefficients, 1, 3, &c. are each of them less than I; so that its convergency is somewhat swifter than by the powers of 80. 8. But another advantage of these new series is, that the numerator and denominator of every term except the first, in each of them, is divisible by 8; in consequence of which, the arithmetical 31.80 39.80' operation by them is much facilitated, the division by 80 being exchanged for a division by 10, which is no more than removing the decimal point. These series, then, when the factors which are common to both numerators and denominators are expunged, will stand as follows (each of which still converging somewhat quicker than by the powers of 80), and we shall have the circumference of a circle whose diameter is 1, Some curious series for the same purpose have been recently investigated by Mr. Wallace of the Royal Military College, for which we must refer this ingenious paper on the subject, in the Edinbarzh Transactions. QUADRATURE, in astronomy, that aspect of the BD when she is 90° distant from the sun; or - she is in a middle point of her orbit, between se points of conjunction and opposition, namely, .: the first and third quarters. See ASTRONOMY. QUADRATURE LINES, are two lines placed on Geater's sector: they are marked with Q. and 5, ,, 8, 9, 10; of which Q. signifies the side of the square, and the other figures the sides of polygons ef 5, 6, 7, &c. sides. S, on the same instrument, stands for the semidiameter of a circle, and 90 for as equal to ninety degrees in circumference. QUADRATUS. See DEPRESSOR LABII INFERIORIS. It is also the generic name of various other muscles. QUADRATUS FEMORIS. (quadratus, from astra, a square; so called from its supposed shape.) A muscle of the thigh situated on the ontide of the pelvis. It is a flat, thin, and Besay muscle, but not of the shape its name world seem to indicate. It is situated immediately below the gemini. It arises tendinous and fleshy from the external surface and lower edge of the tuberosity of the ischium, and is inserted by short tendinous fibres into a ridge which is seen extending from the basis of the trochanter major to that of the trochanter miDor. Its use is to bring the os femoris outwards. Q. GENE. See PLATISMA MYOIDES. LABII INFERIORIS. See DEPRESSOR LABII INFERIORIS. Q. LUMBORUM. Quadratus, seu Lumbaris externus of Winslow. A muscle situated within the cavity of the abdomen. This is a small, flat, and oblong muscle, that has gotten the name of quadratus from its shape, which s that of an irregular square. It is situated aterally, at the lower part of the spine. It aruses tendinous and fleshy, from about two Laches from the posterior part of the spine of the ilum. From this broad origin it ascends obliquely inwards, and is inserted into the nasverse processes of the four superior lumbar vertebræ, into the lower edge of the last rib, and, by a small tendon, that passes up under the diaphragm, into the side of the last vertebra of the back. When this muscle acts singy, it draws the loins to one side; when both muscles act, they serve to support the spine, and perhaps to bend it forwards. In laborious 19.10 2 B 23.10 31.10 39.10' respiration, the quadratus lumborum may assist in pulling down the ribs. See PLATY Q. MAXILLE INFERIORIS. MA MYODES. Q. RADII. See PRONATOR RADII QUA. DRATUS. QUADREL, in building, a kind of artificial stone, so called from its being perfectly square. The quadrels are made of a chalky earth, &c. and dried in the shade for two years. These were formerly in great request among the Italian architects. QUADRENNIAL. a. (quadrennium, Latin.) 1. Comprising four years. 2. Happening once in four years. QUA'DRIBLE. a. (from quadro, Latin.) That may be squared (Derham). QUADRICAPSULAR PERICARP, in botany, having four capsules to a flower: as in rhodiola. QUADRIDENTATE DOWN, in botany. A four-toothed down. Having four teeth on the edge. As in rudbeckia. QUADRIFID CALYX, in botany. A four-cleft perianth: as in rhinanthus.-Quadrifidum folium. A four-cleft leaf. Cut into four segments with linear sinuses, and straight margins. QUADRIGA, in antiquity, a car or chariot drawn by four horses. On the reverses of medals we frequently see the emperor or victory in a quadriga, holding the reins of the horses; whence these coins are, among the curious, call ed nummi quadrigati, and victoriati. QUADRIJUGOUS LEAF, in botany. Pinnate, with four pairs of leaflets. QUADRILATERAL, or QUADRILATERAL FIGURE, is a figure comprehended by four right lines; and having consequently also four angles, for which reason it is otherwise called a quadrangle. The general term quadrilateral comprehends these several particular species or figures, viz. the square, parallelogram, rectangle, rhom bus, rhomboides, and trapezium. If the opposite sides be parallel, the quadrilateral is a parallelogram. If the parallelogram have its angles right ones, it is a rectangle; if oblique, it is an oblique one. The rectangle having all its sides equal, becomes a square; and the oblique parallelogram having all its sides equal, is a rhombus, but if only the opposites be equal, it is a rhomboides. All other forms of the quadrilateral are trapeziums, including all the irregular shapes of it. The sum of all the four angles of any quadrilateral is equal to four right angles. Also, the two opposite angles of a quadrilateral in scribed in a circle taken together, are equal to two right angles. And in this case the rectangle of the two diagonals, is equal to the sum of the two rectangles of the opposite sides. For the properties of the particular species of quadrilaterals, see their respective names, SQUARE, RECTANGLE, PARALLELOGRAM, RHOMBUS, RHOMBOIDES, and TRAPEZI UM. QUADRILATERALNESS. s. The property of having four rightlined sides. QUADRILLE. (quadrillo, Span. a party of four persons.) A game at cards played by four persons, the number of cards being forty; to effect which the tens, nines, and eights are discarded in each suit from the pack. The deal is made by distributing the cards to each player, three at a time, for any two rounds, and once four a time, beginning with the righthand player, who is the elder hand. The stakes, consisting of several equal billets or contracts comprising the counters and fishes, are distributed among the players, who agree upon their value, as also upon the number of tours, which are usually ten. After the trump is named, the right-hand player leads, and if the ombre, either alone or with a friend, win all the tricks, he gains the vole, or if six tricks, the game; but if he get only five tricks he loses by remise, and if only four, loses by codill. The game, consolation, matadores, bastos, and other payments are variously regulated, according as the game is won or lost. The holder of either of the red aces is entitled to a fish for each. The chief rules are as follows: your 1. When you are the ombre, and friend leads from a mat, play your best trump, and then lead the next best the first opportunity. 2. If you possess all the trumps, keep leading them, except you have other certain winning cards. 3. If all the mats be not revealed by the time you have won six tricks, do not risk playing for the vole. 4. When you are the friend called, and hold only a mat, lead it, but if only a mat guarded by a small trump, lead the small one; though when the ombre is last player, lead the best trump you have. 5. Punto in red, or king of trumps in black, are good cards to lead when they are your best, and should either of them succeed, then play a small trump. 6. When the ombre leads to discover the friend, if you hold king, queen, and knave, put on the knave. 7. Preserve the called suit, whether friend or foe. 8. When playing against a lone hand, never lead a king, unless you have the queen, nor change the suit, nor permit, if possible to prevent it, the ombre to be last player. 9. Call on the strongest suits, except you have a queen guarded, and if elder hand you have a better chance than when middle hand. 10. A good player may succeed with a weaker hand when either elder or younger, than if middle hand. better N. B. Spadille and basto are always trumps; by which the red suits have one trump more than the black. There are three matadores, viz. spadille, manille, and basto, which force all inferior trumps, but if an ordinary trump be led, you are not obliged to play a matadore; though if spadille be led, and you hold manille or basto unguarded, you must play it; also if manille be led, and you have basto unguarded, that must be played. 1. The first thing, after you have seen your cards, is, to ask leave; to pass; or play sans prendre; and if you name a wrong trump, you must abide by it. 2. If all the players pass, he who has spadille is obliged to play (called forced spadille;) but if he do not make three tricks, he is not basted. 3. The player, when he calls a king, ought to have a fair probability of winning three tricks, to prevent being basted. 4. Therefore, such games only are set down as give a fair chance to win by calling a king with directions at the end of each what to lead. Calculations for those who have made some progress in the game. 1. What are the odds that my partner holds one out of any two cards? Ans. That he holds one card out of any two tertain cards, is about 5 to 4 in his favour; and if you hold one matadore, those odds are your favour that your partner holds one of be other two, and consequently you may play your game accordingly. Again, suppose you call a king, having a lave and one small card of another suit in your hand, you have 5 to 4 in your favour that your partner holds either the king or queen of Last suit, and consequently you have the odds in your favour to win a trick in the same. 2. What are the odds that my partner holds aze out of any three certain cards? fas. That he holds one out of any three certia cards, is about 5 to 2 in his favour; and if vas have no matadore, but with the assistance of one of them have great odds to win the game; observe, that it is about 5 to 2 that your partBr holds one of them, you having none. This calculation may be applied to many Liber cases. Games in red, calling a king. 1. Spadille, manille, two small hearts or dmonds, the queen of clubs, and one small ene, and four small cards of the other suit. Lead a small trump. 2. Spadille, manille, two small hearts or asionds, with the knave and two small clubs, and three small cards of the other suits. Lead a smail trump. 3. Spadille, manille, two small hearts or damonds, three small clubs, and three small cards of the other suits. Lead a small trump. 4. Spadille, punto, king, queen, and one small heart or diamond, three small clubs, the queen, and one spade. Lead punto. 5. Spadille, punto, king, knave, and one smal heart or diamond, the knave and two Sail clubs, and two small spades. Lead pun to.. 6. Spadille, king, queen, knave, and one small heart or diamond, with the queen, knave, and one small club, and two small spades. Lead the king of trumps. 7. Spadille, three, four, five, and six of berts or diamonds, king of clubs and one ore, queen and two small spades; when you have the lead play a small trump; in the second lead play a spadille. 8. Manille, basto, punto, and two small hearts diamonds, three small clubs, and the knave and one spade. Lead manille. 9. NB: Manille, basto, king, and two small hearts or diamonds, queen and one all club, and three small spades. Lead ma nille. 10. N. B. Manille, basto, queen, and two all hearts or diamonds, queen and two small clubs, knave and one spade. Lead ma 11. Manille, basto, with the three smallest bars or diamonds, queen and one small club, kee and two small spades. Play a small 12. N. B. Manille, punto, king, and two smal beans or diamonds, queen knave, and one small club, king, and one small spadeLead manille. 13. Manille, punto, queen, and two small hearts or diamonds, queen and one small club, king and two small spades. Play a small trump. 14. Manille, punto, and three small hearts or diamonds, knave and one small club, king, queen, and one small spade. Play a small trump. 15. Manille, and the four smallest hearts or diamonds, queen and one small club, king, queen, and one small spade. Play a small trump. 16. N. B. Basto, punto, king, and two small hearts or diamonds, king and queen of clubs, queen and two small spades. Lead basto. 17. N. B. Basto, punto, queen, and two small hearts or diamonds, queen, knave, and one small club, king and queen of spades. Lead basto. 18. N. B. Basto, punto, and three of the smallest hearts or dianionds, king and queen of clubs, queen, knave, and one small spade. Play a small trump. 19. Basto, and the four smallest hearts or diamonds, king and queen of clubs, queen, knave, and one small spade. Play a smal trump. 20. N. B. Punto, king, queen, and two small hearts or diamonds, king and queen of clubs, queen, knave, and one small spade. Lead punto. 21. Punto, king, and three small hearts or diamonds, king and queen of clubs, queen, knave and one small spade. Play a small trump. Games in black, calling a king. 1. Spadille, manille, and two small clubs or spades, the knave and two small hearts, and three small diamonds. Lead a small trump 2. N. B. Spadille, manille, and two small clubs or spades, queen and two small hearts, and three small diamonds. Lead a smal trump, 3. Spadille, manille, and two small clubs or spades, three small hearts, three small diamonds. Lead a small trump. 4. N. B. Spadille, king, queen, and two small clubs or spades, with the queen and one small heart, three small diamonds. Lead the king of trumps. 5. Spadille, king, knave, and two small clubs, queen and two diamonds, two small hearts. Play a small trump. 6. Spadille, queen, and three small clubs or spades, queen and two small hearts, two sinall diamonds. Play a small trump. and 7. Spadille, and the four smallest clubs or spades, king and one small heart, queen two small diamonds. Play a small trump. 8. Manille, basto, king, and two small clubs or spades, three small hearts, and two small diamonds. Lead manille. 9. Manille, basto, queen, and two small clubs or spades, three small hearts, queen and one small diamond. Lead manille. 10. Manille, basto, knave, and two small clubs or spades, knave and one heart, three small diamonds. Lead manille. 11. Manille, basto, and the three small clubs or spades, queen, and two small hearts, knave and one small diamond. Lead manille. 12. N. B. Manille, king, queen, and two small clubs or spades, king and one small heart, queen, knave, and one small diamond. Lead manille. 13. N. B. Manille, king, knave, and two small clubs or spades, king and one small heart, queen and two small diamonds. Lead manille. 14. Manille, king, and three small clubs or spades, queen and two small hearts, king and one small diamond. Play a small trump. 15. Manille, and the four smallest clubs or spades, king, queen, and one small heart, two small diamonds. Play a small trump. 16. N. B. Basto, king, queen, and two small clubs or spades, queen and two small hearts, king and one small diamond. Lead basto. 17. N. B. Basto, king, knave, and two small clubs or spades, knave and one heart, king and two small diamonds. Lead basto. 18. N. B. Basto, king, and three small clubs or spades, king and queen of hearts, queen and two small diamonds. Play a small trump. 19. Basto, and four of the smallest clubs or spades, king and queen of hearts, queen, kuave and one small diamond. Play a small trump. 20. N. B. King, queen, knave, and two small clubs or spades, king and queen of hearts, knave and two small diamonds. Lead the king of trumps. 21. King, queen, seven, six, and five of clubs or spades, king and queen of hearts, queen, knave, and one small diamond. Lead the king of trumps. Those cases, both in red and black, which are marked thus (N. B.) are very good games to play, and you have the odds on your side to win those which are not marked. N. B.-You are to call to your strongest suit, except you have a queen guarded. And if you are elder hand, you have a fairer chance to win the game than if middle, because leading a trump frequently makes your adversaries play against each other. The following are the laws of the game. 1. The cards are to be dealt to the right hand, by fours and threes, and no otherwise; and the dealer is at liberty to begin by four or three; if in dealing there should come one or more faced cards, there must Le a new deal, unless the last card. 2. If there be too many or too few cards in the pack, there must be a new deal. 3. He who deals wrong, is to deal again. 4. He who has asked leave is obliged to play. 5. No one is basted for playing out of turn, but the card played may be called at any time in that deal, provided it do not cause a renounce; or either of the adversaries may demand the partner of him who played out of his turn, or his own partner, to play any suit he thinks fit. 6. The three matadores cannot be forced by an inferior trump, but the superior forces the inferior, when lead by a first player. 7. Whoever names any suit for trumps, must abide by it. 8. If you play with eleven cards, you are basted. 9. If you play sans prendre, or have matadores, you are to demand them before the next dealer has finished, otherwise you lose the benefit. 10. Any person naming his trump without asking leave, is obliged to play sans prendre, unless he is the younger hand, and all the rest have passed. 11. After the game is won, if the person who won the sixth trick play the seventh card, he is obliged to play for the vole. 12. If you have four kings dealt, you are at liberty either to call a queen, to one of your kings, except the queen of trumps; or to call one of your kings. 13. If any person separate a card from the rest, he ought to play it, if the adverse party have seen it, unless he plays sans prendre. 14. If the king call, or his partner play out of his turn, no vole is to be played for. 15. No person is to be basted for a renounce, unless the trick is turned and quitted; and if any person renounce, and be discovered, should the player happen to be basted by such renounce, all the parties are to take up their cards, and play them over again. 16. Forced spadille is not obliged to make three tricks, nor is allowed to play for the vole. 17. Whoever undertakes to play the vole, has the preference of playing before him who offers to play sans prendre only. 18. If all parties agree to it, let the person have the preference of playing who plays for the most tricks, which will prevent small games. 19. The ombre is entitled to know his king called, before he declares for the vole. 20. When six tricks are won, he who won the sixth ought to say, I play the vole; or, I do not play the vole; or, I ask -, and nothing else. 21. He who wins the vole, is to take double the stake played for out of the pool. 22. He who asks leave (if elder hand), may play sans prendre, in preference to any of the. other players. 23. If you have one or more kings, you may call yourself, but must win six tricks. 24. If you play the king surrendered, he must win six tricks who demands the king of any person. 25. He who has passed once (unless he has spadille) has no right to play afterwards; also. |