2. 24000. 16535 65. The average weight of 69 persons is 11 stone; of 70 and the interest on the same sum for the same time is £28 3s. 9d. persons, 11 stone 1 lb. What is the weight of the 70th ? Find the rate per cent., and the sum. 66. To pay a bill of £300 three months before date, a person 84. A person's income is derived from the proceeds of £4550 sells 33 per cent. stock at 90. Discount being allowed at 4 per at a certain rate per cent., and £5420 at 1 per cent. more than cent. per annum, how much stock must he sell, and what does the former. His whole income is £453. Determine the rates. be gain or lose by paying at once ? 85. One clock gains 4 minutes and another loses 4 minutes 67. A shilling weighs 3 dwt. 15 grains, and is ad fine. What in 12 hours. Find the time indicated by each clock, when one is the value of (1) a pound Troy ; (2) a pound avoirdupois of appears to have gained 164 minutes upon the other, supposing pure silver ? them to start together at noon. 68. A person had two-fifths of a coal mine; he sold three 86. The gold coinage of one country contains 1 part silver to fourths of his share, and divided the remainder between his two 11 parts of gold without alloy, that of another 1 part of alloy to sons, giving four-fifths to the elder and £200 to the latter. 23 parts of gold. It is found that 46 of the first weigh as much Find the value of the mine. as 88} of the second. The intrinsic value of silver is sth that 69. By selling tea at 58. 4d. a pound, a grocer clears one of gold. Find the par of exchange. eighth of his outlay; he then raises the price to 6s. What does 87. A man insures his life to the extent of 10 per cent. upon he clear per cent. at the latter price ? his whole income; after deducting this, he pays 8d. in the 70. If a pipe of 6 inches bore discharges a certain quantity of pound income tax on the remainder. His net income is £957. water in 4 hours, in what time will 3 pipes of 1, 2, 3 inches bore Find his gross income. respectively discharge the same quantity, the water flowing in 88. A can do a piece of work in 6 days, which B can destroy each case with the same velocity ? [N.B. The bores of the in 4. A has worked 10 days, during the last 5 of which B has pipes are proportional to the squares of their diameters.] been destroying ; how many days must A now work alone in 71. A piece of work must be finished in 36 days, and 15' men order to complete his task ? are set to do it, working 9 hours a day; but after 24 days it is 89. A and B lay out equal sums in trade. A gains £100, found that only of the work is done. If 3 additional men be and B loses so much that his money is now only of A's. But then put on, how many hours a day will they have to labour in if each gave the other of his present sum, B's loss would be order to finish the work in time? diminished by one-half. What had each at first ? 72. Seven men had water enough for 13 days, allowing 11 90. On certain goods the import duty is 150 per cent. on pints per man daily. After 5 days some water escaped, and one their prime cost. The duty is reduced one-half, but the cost of man died, and the water lasted the 13 days. How much was production increases 10 per cent. Determine what would have lost? been the price of goods sold now at £46 4s., allowing 20 per 73. A has twice as much money as B. They play together cent, profit in each case. for a certain stake. At the end of the first game B wins from A one-third of A's money. What fraction of the sum B now KEY TO EXERCISES IN LESSONS IN ARITHMETIC.-XLVI. has must A win back in the second game, that they may have EXERCISE 64. exactly equal sums? 1. 15. 38. 3011' cubic yds.; 74. If 5 pumps, each having a length of stroke of 3 feet, 23. 7 weeks altogether. working 15 hours a day for 5 days, empty the water out of a 3. 12 stone. 24. £20 6s. 018 d. 39. £20 6s, 3d. mine, what must be the length of stroke of each of the 5 pumps, 4. £286492500. 25. 88 oz. £343 108.3ja. 40. 876, 99.54 yards which, working 10 hours a day for 12 days, would empty the 5. 33} per cent. 26. £47 5s. approximately. same mine, the strokes of the former set of pumps being per 41. 41 of £1215 and 7. 83148.8. formed four times as fast as those of the latter ? 28. 18; £223. $5 of £1245 respec 8. 17317 acres. 29. 895 and 11277, or tively. 75. From 1797 to 1821 cash payments were suspended. 9. 02031. 1611 and 6265. 42. 8853't. Before that time the value of gold was £3 17s. 10d. per oz. ; 10. 15 and 180. 30. The first way is 43. 212123's yards. but in 1815 it rose to £4 13s. 6d. per oz. How much per cent. 11. 914 yds. 1 ft. 9 in. the cheapest. 252 44. £52 10s.; £58 68.8d. had the currency depreciated ? £15681 155. 43 d. quintals. more income. 76. Two clocks, one gaining 3 min. and the other losing 2 12. 18 per cent. 31. £38 Os. 5d. 45. 2.80335 acres. min. a day, are set right at noon. What is the time by the 13. 8d. 32, 3045 sequins. 46. £l lls. 4 d. first clock when the second indicates noon a week afterwards ? 14. £394 Os. 8 d.ş. 33. The second way is 47. 19041 francs 59 120 the eheapest. cents, 77. A trader fits out 4 ships in succession to run a blockade: 15. 114 paper gulden. 16. £375. quintals. he reckons the total outlay on each ship after the first to be 8 48. 372. 17, 352 persons. 34. 2250. [10s. 49. 2.97d. per cent. more than on the one that preceded it. The first and 18. 7-060002. 35. 3 per cent. £54330 50. In 6 years. third get into port, and he gains 160 per cent. on their cost, 19. 139 inches. 36. 16720 tons. 51. The ratio of 3 to 7. while the second and fourth are taken. What is his gain per 20. £1060. [13s. 37. 729, 432, 3348, 27 52. £4 48., £2, £1 165. cent. on the whole ? 21, 65 per cent. £574) respectively. respectively, 78. The price of raw cotton being 5d. a pound, and of cleaned cotton 6d. a pound, how much per cent. in weight must be lost in cleaning, the cost of cleaning being neglected ? LESSONS IN BOTANY.-XXXV. 79. The regulations respecting Great Exhibition tickets, from the opening, on Thursday, May 1, to Saturday, October 18, were SECTION CVI.—THYMELACEÆ, OR DAPHNADS. as follows:-Three guinea season tickets alone admit to the Characteristics : Perianth tubular, petaloid ; stamens perigyopening. £1 was charged on May 2nd and 3rd, and on three nons, their number equal to the divisions of the perianth, occaexceptional days (not in May, nor shilling days). From May 5th sionally donble or fewer; ovary free, uni-locular; ovules pendent; to 17th the charge was 58., and for the rest of the month 28. 6a., fruit drupaceous or in nuts, ordinarily one-seeded, exalbuminous; except one day in each week, when the charge was 58. After stem usually ligneous ; leaves simple. May the charge for admission was 1s. on four days of the week. All the species of the genus Daphne contain an acrid principle, If of the remaining days 18 were 5s. days, and the rest 28. 63. which gives them a vesicating property. The Daphne Fortuni days, estimate the saving, by taking a season ticket, of a person is a very beautiful plant; it was brought from China by Mr. who proposed to be a daily visitor. Fortune, some years ago, and is now cultivated in England. 80. If 5 per cent. be lost by selling a horse for £38, at what This gentleman also introduced the golden-flowered Edgeworthia, price must three others, which cost each the same as the first, or the Edgeworthia chrysantha (Fig. 265). It is a very beautiful be sold, in order to gain 10 per cent. on the whole ? member of the Daphne genus, and must not be confounded 81. What would be the value of 135-74, if the local value of with the Reptonia, which was originally called Edgeworthia. the digits increased eightfold from right to left ? SECTION CVII.-LORANTHACEÆ. 82. How many plots of ground of 33, square yards can be cut Characteristics : Calyx adherent to the ovary ; petals free or from a field containing 4 acres, 3 roods, 9 poles, 191 square coherent, epigynous, four, six, or eight, valvate in wstivation ; yarda, whose breadth is 135 yards ? and what will be the stamens opposite to the petals or to the divisions of the simple width of the remaining strip, after the plots are marked off ? perianth ; ovary nni-locular; ovule pendent; berry one-seeded; 83. The discount on a certain sum for one year is £27 105., | embryo placed at the surface of an abundant fleshy albumen; 6. 512. 27. 9216. small dichotomous shrubs, always parasite; leaves opposite, flowers complete, or polygamous, or diæcious; embryo insepeentire ; flowers sometimes diccious. rable. Members of this natural family inhabit the intertropical re- The most remarkable species of this class is the Rafflesia gions. Their bark contains adhesive material, like birdlime, Arnoldi (Fig. 267), a native of Sumatra, where it grows on the intermediate in its general nature between wax and caoutchouc. trunk of a cissus, and bears a single flower no less than nine. The mistletoe (Viscum album, Fig. 266) is the only species feet in circumference. Its nectary has a capacity of twelve which represents the family in our land. pints, and its weight is not less than fifteen It is a diccious plant, with thick fleshy pounds. Before its expansion the floral leaves, greenish flowers scarcely apparent, bud appears like a great cabbage; the and sessile. The mistletoe was much re bracts soon expand, and the perianth beverenced by the ancient Druids, who attri comes developed. Its fleshy colour and buted to it various mysterious properties. cadaverous odour attract flies and other Even at this day the inhabitants of Java en insects, which are necessary to the process tertain a superstitious respect for the Ficus of its fecundation. This curious member religiosa, upon which an individual of the of the vegetable world has been described natural family Loranthaceve grows. They at length in Vol. I., page 185. 265. EDGEWORTHIA CHRYSAN 268. RAFFLES'S PITCHLE-PLAST THA, 266. THE MISTLE (NEPENTIES TOE (VISCUM ALBUM). SIANA). 1, THE LEAT 267. RAFFLESIA AR 2, THE MALE FLOWER :: NOLDI. 3, THE FRUIT. believe that the shades of SECT. CIX.-NEPENTHA their ancestors wander CEÆ, OR NEPENTHS. under the vaulted canopy Characteristics : Sub-lig formed by these curious neous plants of tropical Asin trees, and are gladdened and Madagascar; flowers by a view of the para in racemes, diæcious ; per sites. rianth "herbaceous, four, partite; stamens sixteen SECT. CVIII.-HYDNORA coherent in a central co CEA, RAFFLESIACEÆ, 267 lumn; ovary free, four, CYTINACEÆ, APODANTHACEÆ, AND BALA celled ; capsule localicidal NOPHORACEÆ. The Nepenthes, type of this family, possesses alternate These five natural orders constitute a group of plants which leaves, the petiole contracted at its base, but further on expanding have been collectively termed Rhizogens or Rhizantheæ from into a flat limb, but its mid-rib is prolonged, and bears a new fosome Greek words which mean "flower-producing roots." They liaceous expansion like a pitcher in form, supplied with a COFER are supposed to constitute an intermediate bat distinct class attached by a kind of hinge, on which it opens and shuts. The between the Phanerogamia and the Cryptogamia. The following pitcher closed at night is open during the day, and secretes are their leading characteristics :-Plants composed of cellular on its interior a fluid, insipid in some species, slightly saccharin tissue, pervaded by a few vessels; parasites upon the roots or in others. The largest and finest species was discovered about stems of other plants ; leaves reduced to mere scales, never thirty years ago at Singapore by Sir Stamford Baffles, and tes deprived of stomata, and vessels, generally imbricate ; received the name of Nepenthes Raflesiana (Fig. 268). EARLY HISTORY LESSONS IN ASTRONOMY.-I. remote than any other science, many important discoveries having been made in it at a period anterior to all written history. OBJECTS AND OF THE SCIENCE EARLY In the book of Job, which is usually admitted to be the most ASTRONOMERS : THALES, HIPPARCHUS, PTOLEMY. ancient book in the world, we find reference to Arcturus, Orion, Or all the sciences which arrest the attention and engage tho and the Pleiades, showing that, even at this early period, names thought of mankind, the science of astronomy is assuredly the had been given to some of the constellations and stars. most grand, the most ennobling, and the most sublime. Most We can easily understand why this should be so. Every one others, though they tend greatly to expand and enrich the mind, of us, when walking alone on a clear night, when the moon has chain it down to the earth; but this lifts it up, and carries it set, and the whole concave of heaven is studded with innumerable away far beyond the boundaries of the finite, till it is almost stars, must have felt an anxious desire to know something of the lost in the illimitable void of space. history and motions of those bodies. This desire was felt in the Astronomy seems to lift man out of himself, and to place him early ages of the world, and in the East, where the science seems on a standpoint far removed from the world he inhabits, which to have had its origin, the settled weather, the clearness of the it reduces to a mere unit in the glorious whole; and as he air, and the cloudlessness of the sky, would all render these beholds the unaltering regularity and unceasing motion of the observations more easy. In those early ages, too, men lived far heavenly bodies by which he is surrounded, and by slow degrees more in the open air than in the present day. Shepherds, for comes to perceive that all their varying and apparently compli- instance, often watched with their flocks during the whole night, cated motions resolve themselves into the most beautiful and thus they would have frequent and favourable opportunities simplicity, and are all governed by a few plain and simple laws, for watching and noticing the movements of the stars. This he is led to see fresh proofs of the power and wisdom of Him occupation, too, would serve well to beguile the otherwise tedious who by His word called them into being, and launched them hours of night, and we find accordingly that shepherds were the forth in space. first astronomers. By this science the student learns the hard lesson that the Another reason why the science attracted so much attention evidence of the senses is not always to be depended on; that the was its great practical importance. Men soon noticed the apparently immovable earth is in reality in a state of continual regular changes of the seasons. They would see that at one motion, both on its own axis and around the sun ; and that the time winter cold seemed to reign over all, and apparent death real movements of the heavenly bodies are quite different from held all the vegetable world. Spring then followed, with its those which are apparent. He learns, too, that this world, which fresh leaves and opening flowers, and summer and autumn with he has always been accustomed to regard as the largest and their fruits and stores of grain ; and they naturally inquired the most important body in existence, is classed among the orbs of reason of all these changes. They would notice likewise that heaven, and even among the smallest of them; that it is, indeed, during the summer months the sun was absent from them for bat a speck in creation, quite invisible from the nearest of those only a short time, and at noon attained a greater height above fixed stars which stud the sky; and thus he is led to feel his own the horizon than he did in the winter months, when the night insignificance. And yet when he finds that the motions of all was long and the hours of daylight but few; and they would these bodies can be accurately determined, that their sizes and thus come to connect the changing seasons with the motions of even their weights can be measured, that some of the elements the sun, which were accordingly noted with greater accuracy. which enter into their composition can be told, and that their In a similar way the changes of the moon, from the first narrow distances can be ascertained, though so great that light, with all crescent of light to the full round orb, and then back again, its spsed, takes thousands of years to cross the chasm that would early be remarked. separates them from us, he sees something of the immense power One other cause for the study of this science is found in man's with which the human mind has been endowed. innate craving for the supernatuml, or something beyond him. As he advances he finds that the sun and moon, which appear self. The apparent immutability of the heavenly bodies, the like small bodies performing their journeys round the earth, are purity of their light, the regularity of their motions, and, above in reality worlds, the former of them greatly exceeding in size all, the mystery which enveloped them, excited his admiration that on which he lives; that the stars, which he has looked upon and reverence; and hence we find that they early became objects 23 mere points in the sky, are in reality suns, with systems of their of worship to the ignorant, and therefore superstitious, people of own revolving around them--the "contres of life and light to that time. The study of their motions was therefore usually myriads of unseen worlds;" and that these suns, with their pursued for some religious or astrological purpose, and the chief attendant worlds, are all revolving in mighty orbits around one astronomers were priests or professional diviners. common centre, and forming one grand cluster. We can easily understand why this was the case. It was seen The telescope still further extends his view, for by its powerful that the succession of the seasons and the alternations of day and aid he discovers here and there faint nebulo, or patches of cloudy night were caused by the motions of the most important of the light, scattered among the stars; and those at length resolve heavenly bodies, and hence it was supposed that all the rest themselves into complete clusters, similar to that which is made exerted their influences over other matters that were going on up of our car and all the other stars around us. But here the in the world, and that by the careful study of their motions and power of his instrument fails him, and the distances and magni. changes future events in the history of men and nations might tude of these systems are such as to baffle all computation or easily be predicted. We find, accordingly, that astrologers were thonght. Man can only stand on the verge of the infinite, and consulted before any great or important work was undertaken, wonder and adore the glories of Him who filleth all space. and their advice was usually very strictly adhered to. We must therefore come to the study of this science with a The question as to what nations first cultivated this science mind specially prepared for the reception of its truths, being cannot be definitely settled. It seems probable, however, that ready, on the one hand, to receive all truths which shall be the earliest systematic observations of the stars were made by shown to be fully supported by careful observations and proof, the Chaldæans. even though they appear to be sometimes almost contrary to the The path of the sun among the fixed stars was very early svidence of the senses; and, on the other hand, to dismiss all discovered, and these stars were arranged into the twelve con. those crude notions previonsly formed in the mind which, upon stellations, known as the signs of the zodiac, long before the consideration, are not found to be supported by facts, and which historical era. Many of the other constellations were also named, tend, therefore, to hinder and mislead us in our inquiries. The but some were afterwards altered by the Greeks and Romans ; importance of this latter point will be clearly seen when we and even in modern days a few additions have been made, as, for notice how, in the early ages of astronomy, all true progress was instance, the Shield of Sobieski and the Heart of Charles I. effectually checked by the firm hold which certain preconceived It must not be supposed that any resemblance can be traced notions had acquired over the human mind; and how, when at between the shape marked out by the stars and the figures they last the fact of the earth's motion was discovered, and the com- are supposed to represent. The original idea seems to have plicated and cumbrous systems previously believed in wore thas been merely to map out the sky into convenient portions for at one stroke swept away, perseontion and opposition of every examination, and at the samo time to immortalise certain real kind Tere heaped upon the men whose intellect had thus solved or mythical heroes ; but as the system became adopted unithe difficulties of ages. versally, it has been retained to the present day, and serves The science of astronomy dates from an antiquity far more as a ready means of distinguishing and registering the stars. VOL. I. 80 The zodiacal signs are sometimes supposed to have been remarkable when we consider the imperfect nature of the instruconnected with the rural occupation of the ancients. The cluster ments he had to employ. He also observed the irregularities in of stars among which the sun was passing in spring was called the rate of the sun's motion, and determined in what part of its the Ram, because at that time the flocks were sent out into the course its speed was greatest, and thus ascertained that, if the fields. The Lion, too, has been considered symbolical of the motion of the sun was uniform, the earth was not situated in intensity and power of the rays of the summer sun. The the centre of its orbit.. Balance tells of the period of equal day and night; the Scorpion Another thing for which the name of Hipparchus is memorof the unhealthiness of autumn; while the Waterbearer and the able is a catalogue of fixed stars, Fishes betoken the rains and floods of winter. which he formed in order that future The names given to these twelve constellations are as fol- astronomers might be able to detect lows :---Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, any alteration in their position or Sagittarius, Capricornus, Aquarius, Pisces. Their order may, number. He appears to have been however, be more easily remembered by the following lines : led to undertake this task by the apThe Ram, the Bull, the heavenly Twins, pearance of a new star, and though the work of carefully ascertaining and Fig. 1. especially with the imperfect instruThe Man that bears the Watering Pot, ments possessed in his time, & work And Fish with glittering scales. requiring great labour and patience, Among the most important of the phenomena of the heavens he persevered, and completed a list which contained 1081 stars. are solar and lunar eclipses, and these of course attracted the In the progress of this work he made one very important disattention of early astronomers, and at length the true cause of covery. On comparing the place assigned by him to a star in them was discovered. A careful record also appears to have the constellation Virgo with that determined by some distin. been kept of them, so that the Saros, or Chaldæan Period, was guished astronomers nearly two hundred years previously, he discovered. This is a period of eighteen years and eleven days, found a difference of two degrees in its longitude. or 223 lunar months, at the expiration of which the moon enters He then made similar comparisons, where it was possible, again upon its former track in the heavens, and thus the same with respect to other stars, and found the same change in their eclipses are, as it were, repeated. position. It was thus evident that all the stars must have The wonder and anxiety with which these remarkable events moved forward, or else that the point from which the measurewere witnessed can easily be imagined, and when the first ments were taken had moved backwards. This phenomenon is astronomer ventured to predict an eclipse, and his prediction known as the Precession of the Equinoxes, and will be fully was found true, he must have been looked upon as little short of explained in its place. The reason of it was discovered by the divine. The first instance we have on record in which this was great Newton. actually done is in the year 610 B.C., when Thales, the father of Another idea for which we are indebted to Hipparchus was astronomy among the Greeks, foretold an eclipse of the sun. that of representing the positions of the stars on an artificial It seems probable, however, that a similar thing had been pre- globe, and of marking the position of places on the terrestrial viously done by the Chaldæans. globe by means of lines of latitude and longitude. With Thales begins the true history of astronomy. The Nicias, one of the followers of Hipparchus, is said to have Greeks were not, however, distinguished by any great proficiency gone even further than his master, and started an hypothesis in the natural sciences. We find here and there shrewd guesses that the apparent changes in the sky were caused by a daily and faint gleams of truth; but it is always mixed up with revolution of the earth. The idea, however, was not supported fanciful speculations, instead of being supported by careful by any arguments, and was lost sight of for ages. observation and reasoning. They seem, for the most part, to The only other one of the ancient astronomers we shall refer have started with certain principles (which had no existence to in this sketch is Ptolemy, who was a very learned scholar, except in their imaginations), as, for instance, that the earth not only in astronomy, but in mathematics and geography. must be in the centre of the universe, and that, since the circle Having carefully examined the observations of Hipparchus and was the perfection of shape, all the motions of the heavenly others, he at length promulgated' a system known as the bodies must be in circles. They then observed the phenomena Ptolemaic, which, though since proved to be quite erroneous, of the sky, and the apparent motions of the sun and stars, and accounted so well for all known phenomena that its errors could formed cumbrous and complicated systems to try and reconcile not, with the instruments then in use, be discovered; and these appearances with their theories. accordingly, it was universally received until the age of Hence we find all the involved mysteries of transparent wheels Copernicus, and even then it was long before it was entirely revolving one within the other, and carrying with them the given up. planets and stars, of cycles and epicycles, and of crystal spheres According to this system, the earth was the centre, with the in ceaseless rotation, which the followers of Ptolemy were ever planets revolving round it in the following order : The Moon, planning and altering. The true law of discovery--which is to Mercury, Venus, the San, Mars, Jupiter, Saturn, and, beyond all, make accurate observations first, and afterwards start a theory the fixed stars. To account for the apparent irregularities in to explain the appearances--seems to have been quite lost sight their motions, he introduced what he termed epicycles, which of, and hence confusion prevailed. will be understood by reference to Fig. 1. E represents the We must, however, just glance at a few of the names which earth, and A B C the orbit in which the planet should move ; stand prominently forward in the history of the science. but, instead of this, he supposed that there was a point c Anaxagoras and Pythagoras were two of the Greek philosophers moving in this orbit, and that the planet p moved ronnd this who succeeded Thales, and they appear to have had much more point in a small circular orbit or epicycle. The combination of accurate views than most in their day. The truth of the earth's these two motions explained the irregularities. This system motion round the sun seems to have been realised by them, was afterwards rendered much more complicated by the ultersthough it does not appear to have been received by others, and tions introduced by his successors. was opposed by those in power as being impious. The former of the two was indeed sentenced to death on account of his philosophical views; but his sentence was afterwards, through LESSONS IN ALGEBRA.-XIV. the influence of a friend, commuted into banishment for life. The next we notice was the greatest of all the ancient astro SIMPLE EQUATIONS. nomers, Hipparchus, who lived in the second century before 151. Most of the investigations in algebra are carried on by the Christian era. He gave up all attempts to frame a system means of equations. In the solution of problems, for example, for the universe, and occupied himself by carefully watching and we represent the unknown quantity, or numbers sought, by : recording the motions of the sun and planets. The movements certain letter; and then, in order to ascertain the value of this of the sun in particular occupied bis persevering attention, and unknown quantity or letter, we form an algebraic expression from in this way he made a very near approximation to the true the conditions of the question, which is equal to some given length of the year; and the accuracy of his observations is very quantity or number. EXAMPLE. A drover bought an equal number of sheephamed alike they must be united in one, by the rules for reduction in 155. When several terms on the same side of an equation are cows for 840 crowns. 2 athe sheep, aad 12 crowns a-head for the cows. How many did he buy of addition. each? EXAMPLE.--Reduce the equation * + 5b - 4h = 76. Here, let x= the number bought of each; Here, transposing 5b-4h, we have x = 76--56 + 4h; Then 2x = the cost of the sheep in crowns; And uniting 76 - 5b in one term, we have a= 26 + 4h. Ans. And 120 = the cost of the cows in crowns. 156. The unknown quantity must also be transposed, whenHence, 22 + 12x = 840 by the conditions of tho question. ever it is on both sides of the equation. It is not material on Therefore, 14x = 840 by addition; which side it is finally placed, though it is generally brought to And x = 60, the number bought of each. the left-hand side. Here, the last expression is obtained from the preceding one EXAMPLE.—Reduce the equation 2x + 2h = h + d + 3x. by dividing each member by 14, the co-efficient of 14w. Here, by transposition, we have 2h-h-d=3x — 22 It will be perceived, in this example, that the unknown And by incorporation (Art. 155] h-d=l. Ans. quantity, or number sought, is represented by the letter sc; and 157. When the same term, with the same sign, is on opposite from the conditions of the problem, we obtain the quantity sides of the equation, instead of transposing, we may expunge it 142, which is equal to the given quantity 840 crowns. This from each. For this is only subtracting the same quantity from whole algebraic expression, 14c = 840 crowns, is called an equal quantities. equation. 152. An EQUATION, therefore, is a proposition expressing in EXAMPLE.Reduce the equation +3h+d=5+3h+7d. algebraic characters the equality between one quantity or set of Here, by expunging 3h, we have 2+d=b+7d; quantities and another, or between diferent expressions for the And by transposition and incorporation a= 6 + 6d. Ans. same quantity, 158. As all the terms of an equation may be transposed, or This equality is denoted by the sign =, which is read “is supposed to be transposed, and it is immaterial which member equal to." Thus + a=b+c, and 5+ 8 =17 -- 4, are is written first, it is evident that the signs of all the terms may equations, in one of which the sum of w and a is equal to the be changed, on both sides, without affecting the equality. sum of b and c; and in the other, the sum of 5 and 8 is equal to Thus, if we have *-b=d-a; the difference of 17 and 4. Then by transposition, we have --d+a=-+b. The quantities on the two sides of the sign = are called Or, by changing the places of the members, ---+b=-d+a, xembers of the equation; the several terms on the left con 159. If all the terms on one side of an equation be transposed. stituting the first member, and those on the right the second each member will be equal to 0. member. Thus, if &+b = d, then it is evident that a +b-d=0. When the unknown quantity is of the first power, the proposition is called a simple equation, or an equation of the first degree. EXERCISE 25. 153. The reduction of an equation consists in bringing the un 1. Reduce a + 2 - 8 = b - *+* + a. known quantity by itself to one side of the sign of equality, and all 2. Reduce y + ab - hm = a + 2y - ab + hm, the known quantities to the other side, without destroying the 3. Reduce h + 30 + 7x=8-6h + 6x-d+b. equality of the members. 4. Reduce bh + 21 - 4x + d = 12 - 3x + d-701, 5. Reduce 5x + 10 + a= 25 + 4x + c. To effect this, it is evident that one of the members must be 6. Rednce 5c + 25 + 12 - 3 = x + 20 + 50. as much increased or diminished as the other. If a quantity be 7. Reduce a + b - 3x = 20 + - 4x +0. added to one, and not to the other, the equality will be de 8. Reduce x + 3 - 20 - = 34 + 3 - 4 - 54. stroyed. But the members will remain equal 9. Reduce 4 - 2 + 18 = 5x + 8. (1.) If the same or equal quantities be added to each. Ax. 1. 10. Reduce 24 - 24 = 3-8 + 2. (2.) If the same or equal quantities be subtracted from each. 11. Reduce 3 + 52 - 18 = 6x - 22. Az. 2. 12. Reduce 10x + 60 + 7 = 28x + 64 - 12x, (3.) If each be multiplied by the same or equal quantities. 13. Reduce y - 10 - b= 6 - b. 14. Reduce x - 10 +c-14 - = 0. (4.) If each be divided by the same or equal quantities. Ax. 4. The principal reductions in simple equations are those which KEY TO EXERCISES IN LESSONS IN ALGEBRA, are effected by transposition, multiplication, and division. EXERCISE 23. REDUCTION OF EQUATIONS BY TRANSPOSITION. abx + # 1 7. In the equation «-7=9, the number 7 being connected 3aby - 3y. 2xy with the unknown quantity by the sign, the one is subtracted from the other. To reduce the equation, let the 7 be ad 2. al - amy +h-my 5. 40bd + 4cde added to both sides. It then becomes x-7+7= 9+ 7. 12 The equality of the members here is preserved, because one is increased as much as the other. But on one side we have 4dy Sac + 3 -7 and +7. As these are equal, and have contrary signs, they - bh balance each other, and may be cancelled. The equation will EXERCISE 24. then be x = 9+7. 10. 3cx. Here the value of x is found. It is shown to be equal to 1. 19. 9 + 7, that is, to 16. The equation is therefore reduced. The 4ery unknown quantity is on one side by itself, and all the known 20. quantities on the other side. In the same manner, if ab=a; Adding b to both sides, we have ---+b=a+b; Зас ham +Gma 13. And cancelling as before, we have x = a +b. Ans. @ +1- 4a + 60 + 20 22. 154, When known quantities, therefore, are connected with the unknown quantity by the sign + or, the equation is reduced ab + b 2 + xy + bx by TRANSPOSING the known quantities to the other side, and pre 23. 15. cạy rizing the omtrary sign. 4x + 2mx This is called reducing an equation by addition or subtraction, a’ (23 – 9%) Baby ] 6x2 - 2-1 16. because it is, in effect, adding or subtracting certain quantities +a'a'-a tas to or from each of the members. 12 y3 + 2y 17. 25, EXAMPLE.-Reduce the equation a +36 ---m=h-d. x + y * + 2a Here, transposing + 36, we have -m=h--d-- 36; Ax. 3. 1. xy + dy 5dr 8 8. am 1 S. 6. 10. ah a3 + axa 11. 2x + 3 ab 2. 10 - y 26. x = h --d-36 + m. Ans. 9. 1. And transposing-, 18. |