which, if once allowed, would not only have made the clergy to be the exact opposite of what Henry had looked for in him. quite independent, but would have given them the opportunity The case which induoed the king to try conclusions with and the means of wholly subverting the kingly power. They Becket and the clerical party was an exceedingly gross one. A said that if a man contracted with another to do a thing, and priest in Worcestershire had violated a gentleman's daughter, confirmed his promise by an oath, the fact that the oath was and afterwards murdered her father. When the scoundrel was binding only on the conscience gave them jurisdiction, and in about to be brought to trial before the king's justices, Becket this way they drew before the spiritual courts many questions claimed him as a clerk, and getting possession of him, degraded of ordinary contracts, disputes about which onght rightly to have him from his priest's office, and then insisted that he could not been tried in the king's courts of law, which were open to all be tried again in the king's court for the same offence. comers, and from which an appeal lay to the king himself. The These were the circumstances under which King Henry sum. last and most important of t'ie clerical claims, however, was moned the bishops to Westminster; and the meaning of the that which asserted that no clergyman could be brought to trial words "saving our own order" is sufficiently clear. Henry left in the king's courts, civil or criminal, for any breach of agree the ball in a rage, but it was not an impotent one. By promises, ment, however gross, or for any crime, however heinous. If a by threats, by various means, he detached most of the prelates elerk was accused of crime, and was arraigned before the king's from their primate, and he won over the Archbishop of York by judges, the bishop of the diocese in which the prisoner dwelt significant hints about the next incumbent of the see of Cantersent an order to the judge, notifying him that the man was in bury. Last to give in was Becket, who yielded only to the orders, and requiring him to surrender the fellow to the bishop's universal pressure brought to bear upon him, and repented as officer. When brought before the spiritual court the prisoner soon as he had assented. But repentance or no repentance, he was often allowed to clear himself on his simple oath, uncorro- did assent, and with the rest of the prelates professed his borated by any witness, to the effect that he had not done that willingness to observe " the ancient customs of the kingdom"of which he was accused. If he confessed, or if the case was which did not recognise the clerical claims--and to withdraw clearly proved against him, he was put to penance, sometimes the saving clause. he was put in prison, and sometimes—but rarely-he was Henry knew with whom he had to deal. He knew that a degraded from his ecclesiastical rank. In this way crimes of confession of this sort was quite useless unless it could be the most abominable kind, and which, if committed by laymen, embodied in some visible instrument. Taking advantage of his were punishable with death, were done with comparative im- success, of the schism in the Papacy (there were at this time two punity when clerks were the offenders. Nor was this all. By Popos, one at Rome, the other in France, and Henry played off means of an absurd test, persons who were not, nor ever meant one against the other), and of the resolute support of the barons, to be, in holy orders, were admitted to the “benefit of clergy.” who were only too glad to give the spiritual lords a kick down, Ability to read or write, no matter how imperfectly, was taken Henry summoned the primate and all the bishops to meet him to be of itself sufficient proof that a man was a clerk, so that a at Clarendon, a village in Wiltshire, and there, being backed, layman arraigned before the king's justices had only to show like Stephen de Langton on a later occasion, by " the whole that he could read or write what was afterwards appropriately nobility of England," he required their sworn assent to what called "the neck verse," and he was forthwith handed over to have been called the Constitutions of Clarendon. the ordinary to be put to his purgation in the ecclesiastical The “Constitutions” were dreadfully hard eating for the court. bishops, divesting them as they did of nearly all their invidious This monstrous immunity, with its yet more monstrous privileges, some of which it must be confessed were sanctioned abuses, was like the last straw that broke the camel's back. So by those "ancient customs” which the king had sworn the flagrantly unjust was it, both in principle and practice, that all bishops to observe. Suits concerning advowsons and rights of honest men were indignant, and cried aloud for some check presentation were to be decided in the civil courts; no clerk, no upon it. The king, who was by means of it and the other pre- matter of what rank, was to quit the kingdom without the royal tended rights of the clergy gradually ceasing to be master in permission; the pretended right to try questions of contracts his own dominions, resolved to apply a curb, and to wipe away made on oath was to be renounced; excommunicated persons the scandal. From the time when he mounted the throne in were not to be made to find security for their residence in any 1154 he ha striven to restrain the power of the clergy, and, appointed place; laymen were not to be tried in spiritual courts aided by the clear head and bold hand of his bosom friend except by approved good witnesses; no chief tenant of the Thomas à Becket, had striven not unsuccessfully. Great had crown to be excommunicated without the king's assent; the been the wrath poured on Becket's head when, as Lord Chan final appeal in all spiritual causes to be in the king; prelates to cellor of England, he had made havoc altogether of many a pet be regarded as barons of the realm, and to be taxed accordingly; clerical abuse. Under the idea that he would continue the same bishops not to be elected without the royal assent; the privilege policy in a sphere where that policy would have the largest of sanctuary to be curtailed ; and clerks accused of any crime possible scope, Henry offered Becket the archbishopric of Can- to be tried in the king's courts, like other men. terbury when that see was vacant in 1161. Becket, it must in The Great Council of the barons unanimously approved the fairness be admitted, was very averse to accept the offer, and for Constitutions, and, sour as the food was, all the prelates, except thirteen months held out a persistent refusal. Finally, how the primate, swore to accept it “legally, with good faith, and ever, he yielded to the earnest solicitations and orders of the without fraud or reserve." Becket was resolute, though alone; king, and was duly installed as Primato at Canterbury. friends as well as foes besieged his constancy, still he held ont; To the surprise of all men, and to the infinite disgust of the and it was not till Richard de Hastings, Grand Prior of the king, Becket from the day of his consecration pursued a totally Templars, a man who seldom bent his knee, even in prayer, new course to that he had formerly taken. Nowhere was there went down on his knees and besought him, that he gave in. so bold an asserter of clerical rights, nowhere a more untiring Unwillingly, and in hope of getting the Pope to annul his worker on behalf of the power of the Church. He claimed lands oath, he swore like the rest to accept the Constitutions" with which had once belonged to the see of Canterbury, but which good faith, and without fraud or reserve.” had long been independent and in laymen's hands; he excom. Pope Alexander refused to ratify the treaty; he released all municated* the owner of an advowson for ejecting a priest who had sworn from their oaths, and threatened to cxcommuniwho had been presented by himself ; he asserted the right cate everybody who should try to support the king's demands. of the spiritual courts to inquire into questions of contract A long trial of strength ensued. Becket got over to France, confirmed by oath ; and in every respect he proved himself and plotted there against his former friend; Henry took the revenues of the hostile bishops into his own hands, and by dint of perseverance managed to keep the clergy in check; and it is * Excommunication was the expulsion of a man, by the highest probable he would have done very much more than he did had ecclesiastical anthority, from the communion of Christian men. The it not been for the brutal murder of Thomas à Becket, which rights and comforts of the Church were refused to the excommuni. was a blunder as well as a crime. cated; the sacraments were not allowed to be administered to him ; he was reckoned accursed ; and, in times of superstition, he was sup. In the autumn of 1170 Becket had returned to Canterbury, posed to be eternally lost if he died without absolution. Excommuni nominally reconciled to the king ; but the old question--which eation was the great weapon of ecclesiastics, and it was a powerful one should be the greater-being revived, Henry is reported to have in the age of ignorance and moral darkness. said in a hasty moment, " Is there not one of those who eat my bread that will rid me of this trouble ?" To Canterbury with 3. You must stop only as long as you can count one, two, their followers went four knights of Henry's court, and, acting three, four. entirely on their own responsibility, slew the archbishop on the 4. You must pronounce the word which is immediately before steps of the altar. a period, with the falling inflection of the voice. The outery raised in England, where the archbishop was 5. The falling inflection (or bending) of the voice is commonly looked upon with favour, not only on account of his bold marked by the grave accent, thus, '. conduct in standing up for his order, but also because he was Examples. supposed to be the champion of the Anglo-Saxon against the Charles has bought a new håt. Norman Englishman, was loud and sincere. Abroad, the feeling I have lost my gloves. of grief was more than equalled by anger, and a sort of holy Exercise and temperance strengthen the constitution. horror was felt at the bare notion of slaying an archbishop. A wise son makes a glad fåther, King Henry, there is every reason to think, was genuinely sorry The fear of the Lord is the beginning of wisdom. for the violence that had been done. Though his “guide and II. THE NOTE OF INTERROGATION. his companion, and his own familiar friend” had proved to be the sharpest thorn in his side, he remembered too well the ? former days to wish him any personal harm. Notwithstanding, 6. The note or mark of Interrogation is a round dot with a book on him was charged the whole guilt of the murder. Penance above it, which is always put at the end of a question. the most severe, disclaimers the most solemn, and ceremonies 7. In reading, when you come to a note of interrogation, you the most humiliating scarcely served to clear him. Purposely must stop as if you waited for an answer. the Papal Court, which saw in Henry the strongest opponent of 8. You must stop only as long as you do at the period. its pretensions, availed itself of the handle given to it, and 9. You must in most cas pronounce the word which is strove to crush the king under a load of obloqny. To a very placed immediately before a note of interrogation, with the great extent it succeeded. Never again did Henry appear as rising inflection of the voice. the same strong champion of State rights as when he forced an 10. The rising inflection of the voice is commonly marked by aasent to the Constitutions of Clarendon. The ghost of Thomas the acute accent, thus, ': à Bocket, now St. Thomas of Canterbury, haunted him, and the Examples. dend man's hand deprived the conqueror of his victory. Has Charles bought a new hát ? The Constitutions of Clarendon were disregarded, the death Have you lost your glóres ? Hast thou an arm like Gud? of Becket making it impossible for the king to fly in the face of Canst thou thunder with a voice like him? the papal veto upon them. Some little submission of the clerical If his son ask bread, will he give him a stone ? to the kingly power was made, but the work marked out by If he ask a fish, will he give him a sérpent? Henry II., the entire subjection of the clergy to the head of the state, was left unaccomplished till the dawn of the Reformation 11. In general, read declaratory sentences or statements with in England, when it was renewed and carried out in the fullest the falling inflection, and interrogativo sentences or questiong possible manner by that“ stately lord who broke the bonds of with the rising inflection of the voice. Pome," and who was saved by natural causes from committing, Examples. in the case of Cardinal Wolsey, the egregious blunder committed Interrogative. Has John arrived ? by the knights of Henry II. when they plunged their swords Declaratory. John has arrived. into the bosom of Thomas à Becket at Canterbury. Interrogative. Is your father well ? Declaratory. My father is wall. Declaratory. Unto Cæsar shalt thou gò. Maud, daughter of Henry I., was the fifth King of England 12. Sometimes the sentence which ends with a note of interafter the Conquest, and the first of the Plantagenet dynasty. rogation should be read with the falling inflection of the voice. Born at Mans, Normandy 1133, Murder of Thomas à Becket E.camples. Succession secured to Henry Dec. 30, 1170 What o'clock is it? How do you do to-dày? 1176 How much did he give for his book ? Becket made Archbishop of Subjugation of Ireland by Where is Abel thy brother? Canterbury 1163 Henry 1172-5 How long, ye simple ones, will ye love simplicity ? Where wast thou, when I laid the foundations of the earth? July 6, 1189 Sometimes the first part of an interrogative sentence should SOVEREIGNS CONTEMPORARY WITH HENRY II. be read with the rising inflection of the voice, and the last part Denmark, Kings of Norway, Kings of. Scotland, Kings of. with the falling inflection. These parts are generally separated . 1147 Sigurd III. 1131 Malcolm IV.. by a Comma, thus, , Waldemar the Gt. 1157 1153 Magnus V. 1164 14. At the comma, the rising inflection is used, and at the Canute VI. William 1165 note of interrogation the falling inflection. Examples. Alfonso I. 1139 Alfonso VIII. 1126 Shall I give you a péach, or an apple ? Sancho III. 1157 Are you going home, or to school? Alfonso IX. Last Sabbath, did you go to church, or did you stay at hòme ? France, Kings of. Whether is it easier to say, Thy sins are forgiven, or to say, Ariss Sweden, Kings of. and walk ? 1159 Philip II.. 1180 Why did the heathen ráge, and the people imagine vain things ? Lucius III. 1181 Swerker I. 1129 Gereany, Emperor of. Urban III. 1185 Eric I.. Is your father well, the old man of whom ye spake ? 1155 Frederick BarGregory VIII. Charles VII.. 1161 15. Sometimes the first part of an interrogative sentence 1152 Clement III.. 1188 Canute. 1167 must be read with the falling inflection of the voice, and the last part with the rising inflection, Examples. Where have you been to-dày ? At home? Who told you to return? Your fáther? What is that on the top of the house? A bírd ? Is not the life more than meat ? and the body than raímept? 1. THE Period is a round dot or mark which is always put at the What weut ye out to see ? A man clothed in soft raiment ? end of a sentence. What went ye out to see ? A próphet? 2. In reading, when you come to a period, you must stop as How often shall my brother sin against me and I forgive him if you had nothing more to read. Until seven times ? . Csante V. . . Hannel L. . . isaae II. . • 1158 Louis VII. • 1137 • 1187 Larossa . 16. In the following exercises some of the sentences are in this room? How negligent some of our fellow-pupils are ! Ah! questions requiring the rising, and some the falling inflection I am afraid many will regret that they have not improved their time! of the voice. A few sentences also ending with a period are Why, here comes Charles ! Did you think that he would return inserted. No directions are given to the pupil with regard to So soon? I suspect that he has not been pleased with his visit. Have you, Charles ? And were your friends glad to see you? When the manner of reading them, it being desirable that his own is cousin Jane to be married ? Will she make us a visit before she is understanding, under the guidance of nature alone, should direct married? Or will she wait until she has changed her name ? him. But it may be observed that questions which can be My dear Edward, how happy I am to see you! I heard of your answered by yes or no, generally require the rising inflection of approaching happiness with the highest pleasure. How does R060 the voice; and that questions which cannot be answered by yes do? And how is our whimsical old friend the Baron? You must or no, generally require the falling inflection. be patient and answer all my questions. I have many inquiries to make. EXERCISE 1. The first dawn of morning found Waverley on the esplanade in John, where have you beon this morning ? front of the old Gothic gate of the castle. But he paced it long before Have you seen my father to-day? the drawbridge was lowered. He produced his order to the sergeant What excuse have you for coming late this morning? Did you not of the guard, and was admitted. The place of his friend's confinement know that it is past the school hour? was a gloomy apartment in the central part of the castle. If you are so inattentive to your lessons, do you think that you Do you expect to be as high in your class as your brother? Did will make much improvement ? you recite your lessons as well as he did ? No. Lazy boy! Care less child! You have been playing these two hours. You have paid Will you go, or stay? Will you ride, or walk ? no attention to your lessons. Shad you go to-day, or to-morrow? You cannot say a word of them. How Did he resemble his father, or his mother? foolish you have been ! What a waste of time and talents you have Is this book yours, or mine? His, or hers? made ! LESSONS IN GEOMETRY.-II. DEFINITIONS (continued). Who hath believed our report? To whom hath the arm of the 9. An angle is the inclination of two straight lines to each Lord been revealed ? other, which meet in a point, and are not in the same direction. The point in which they meet is called the vertex of the angle, III. THE NOTE OF EXCLAMATION. and each of the two straight lines is called a sicke or leg of the ! angle. The angle itself is generally called a plain rectilineal 17. The note or mark of Exclamation is a round dot with an angle, because it necessarily lies in a plain, and is formed of upright dash or stroke above it, which is always put at the end of straight lines. Curvilineal angles are such as are formed on the a sentence expressing surprise, astonishment, wonder, or admira- surface of a sphere or globe; but the consideration of such tion, or other strong feelings. angles belongs to the higher geometry. The magnitudes of 18. In reading, when you come to a note of exclamation, angles do not depend on the lengths of their legs or sides, but you must stop in the same manner as if it were a note of on the degree or amount of aperture between them, taken at the interrogation. same distance from the vertex. 19. You must stop only as long as you do at a period. An angle is generally represented by three letters, one of 20. You must generally pronounce the word which comes which is always placed at the vertex, to distinguish it particularly immediately before a note of exclamation with the falling inflec- from every other angle in a given figure, and the other two are tion of the voice. placed somewhere on the legs of the angle, but generally at Examples. their extremities; and in reading or in speaking of the angle, the How cold it is to dày! letter at the vertex is always placed between the other two, What a beautiful house that is ! and uttered or written accordingly. Thus, in Fig. 4, which How brightly the sun shines ! represents an angle, the name of the angle is either B A C or How mysterious are the ways of God! How are the mighty fallen in the midst of the battle! CAB: the point A is called its vertex; and the straight lines How are the mighty fallen, and the weapons of war perished ! B A, C A, its sides or legs. Would God I had died for thee, 0 Absalom, my son, my sòn! 10. Angles are divided into two kinds, right and oblique, and Oh, what a fall was there, my countrymen! oblique angles are divided into two species, acute and obtuse. It is a dread and awful thing to die! When one straight line meets another, at any point between Oh! deep enchanting prelude to repose ! its extremities, and makes the adjacent or contiguous angles The dawn of bliss the twilight of our woes! equal to each other, each of them is called a right angle, and the Lovely art thou, O Pèace! and lovely are thy children ; and lovely legs of each of these angles are said to be perpendicular to one are thy footsteps in the green valleys ! another. Thus, in Fig. 5, the straight line A B meets the 21. In onr remarks on the period, the student was taught straight line c d in the point A, and makes the adjacent angles that when he comes to a period, he must stop, as if he had CA B, D A B, equal to each other ; each of these angles is therenothing more to read. At the end of a paragraph, whether the fore called a right angle ; and the straight line A B is said to be period or any other mark be used, a longer pause should be perpendicular to the straight line A c, or D A, and consequently made than at the end of an ordinary sentence. The notes of A c or A D is perpendicular to A B. interrogation and exclamation generally require pauses of the When one straight line meets another, at any point between same length with the period. its extremities, and makes the adjacent angles unequal to each It may here be remarked, that good readers always make their other, each of them is called an oblique angle ; that which is pauses long; but whatever be the length of the pause, the pupil greater than a right angle is called an obtuse angle ; and that must be careful that every pause which he makes shall be a total cessation of the voice. EXERCISE 2. A good scholar is known by his obedience to the rules of the school. Fig. 4. time of school is always punctual. He is remarkable for his diligence Fig. 5. and attention. He reads" no other book than that which he is desired which is less than a right angle is called an acute angle. Thus, to read by his master. appointed for the day. He takes no toys from his pocket to amuse in Fig. 6, the straight line A B meets the straight line o D in himself or others. He pays no regard to those who attempt to divert the point A, and makes the adjacent angles unequal to each Do you know who is a good scholar? Can you point out many the angle c A B, which is greater than a right angle, is called other ; each of these angles is therefore called an oblique angle; A D obtuse ; and the angle D A B, which is less than a right angle, aequus, equal, and latus, a side); isosceles (Greek, 1908, equal, is called acute. and skelos, & leg); and scalene (Greek, skalēnos, unequal), 11. A plane figure, in geometry, is a portion of a plane surface, right-angled, obtuse-angled, and acute-angled. inclosed by one or more lines or boundaries. The sum all 19. An equilateral (equal-sided) triangle is that which has the bonndaries is called the perimeter of the figure, and the por- three equal sides (Fig. 8). tion of surface contained within the perimeter is called its area. 20. An isosceles (equal-legged) triangle is that which has only 12. A circle is a plane figure contained or bounded by a two equal sides (Fig. 9). cured line, called the circumference or periphery, which is such 21. A scalene (unequal) triangle is that which has all its that all straight lines drawn from a certain point within the sides unequal (Fig. 10). figure to the circumference are equal to each other. This point 22. A right-angled triangle is that which has one of its angles a right angle (Fig. 11), in which the anglo at A is the right D Fig. 14, Fig. 15. Fig. 16. Fig. 6. Fig. 7. angle. Tho 'side opposite to the right angle is called the is called the centre of the circle, and each of the straight lines is hypotenuse (the subtense, or line stretched under the right called a radius of the circle. The straight line drawn through angle), and the other two sides are called the base and the per. the centre and terminatod at both ends in the circumference, is pendicular ; the two latter being interchangeable according to called the diameter of the circle. the position of the triangle. It is plain, from the definition, that all the radii must be 23. An obtuse-angled triangle is that which has one of its equal to each other, that all the diameters must be equal to angles an obtuse angle (Fig. 10). each other, and that the diameter is always double the radius. 24. An acute-angled triangle is that which has all its angles In speaking or writing, the circle is usually denoted by three acute; Figs. 8 and 9 are examples as to the angles, but there letters, placed at any distance from each other, around the is no restriction as to the sides. circumference ; thus, in Fig. 7, the circle is denoted by the of its angles perpendicular to the opposite side, or to that side In any triangle, a straight line drawn from the vertex of one letters ACB, or AEB; or by any three of the other letters on the circumference. The point o is the centre ; each of the produced (that is, extended beyond either of its extremities in straight lines o A, O B, O C, O E, is a radius, and the straight a continued straight line), is called the perpendicular of the line A B is , diameter. triangle ; as in Fig. 12, where the dotted line A D is the perpen13. An art of a circle is any part of its circumference; the dicular of the triangle A B C; and in Fig. 13, where the dotted chord of an are is the straight line which joins its extremities. line G drawn from the point o to the dotted part of the base produced is the perpendicular of the triangle E F G. 25. A quadrilateral figure, or quadrangle, is a plane rectilineal 14. A segment of a circle is the surface inclosed by an aro and its chord. Fig. 17. Fig. 19. figure contained by four straight lines, called its sides. The Thus, in Fig. 7, the portion of the circumference Auc, straight line which joins the vertices of any two of its opposite whose extremities are a and c, is an arc; and the remaining angles, is called its diagonal. Quadrangles are divided into portion A B C, having the same extremities, is also an arc ; the various kinds, according to the relation of their sides and straight line Ac is the chord of either of these arcs. The sur angles; as parallelograms, including the rectangle, the square, face included between the arc A M C and its chord A C, is the the rhombus, and the rhomboid ; and trapeziums, including the Bezment Arc; there is also the segment A B C. The surface trapezoid. included between the radii o C, O B, and the arc c B, is called 26. A parallelogram is a plano quadrilateral figure, whose the sector c O B; the remaining portion of the circle is also a opposite sides are parallel; thus, Fig. 14, A C B D, is a parallelosector. gram, and A B, C D, are its diagonals. 16. A semicircle is the segment whose chord is a diameter. 27. A rectangle is a parallelogram, whose angles are right Thus, in Fig. 7, ACB or I B is a semicircle. The term angles (Fig. 15). semicircle, which literally means half a circle, is restricted in 28. A square is a rectangle, whose sides are all equal (Fig. 16). D Fig. 20. Fig. 21. Fig. 22. Fig. 13. 29. A rhomboid is a parallelogram, whose angles are oblique. geometry to the segment thus described; but there are many The opposite angles of a rhomboid are equal to one another other ways of obtaining half a cirole. (Fig. 14). 17. Plane rectilineal figures are described under various 30. A rhombus, or lozenge, is a rhomboid, whose sides are all beads; as trilateral or triangular; quadrilateral or quadrangular; equal (Fig. 17). and multilateral or polygonal. 31. A trapezium is a plane quadrilateral figure, whose oppo. 18. A triangle (Figs. 8, 9, 10, and 11) is a plane rectilineal site sides are not parallel (Fig. 18). figure contained by three straight lines, which are called its rider. No figure can be formed of two straight lines ; hence, of its sides parallel (Fig. 19). 32. A trapezoid is a plane quadrilateral figure, which has two an angle is not a figure, its legs being unlimited as to length. 33. A multilateral figure, or polygon, is a plane rectilineal Triangles are divided into various kinds, according to the figure, of any number of sides. The term is generally applied relation of their sides or of their angles : as equilateral (Latin. I to any figure whose sides exceed four in number. Polygons are divided into regular and irregular; the former having all their line and 6 in the left-hand line stand in lines which meet in a sides and angles equal to each other; and the latter having any square containing 24, which is therefore the product of 4 multivariation whatever in these respects. The sum of all the sides plied by 6. of a polygon is called its perimeter, and when viewed in position It may be observed that 6 in the top line and 4 in the leftits contour. Irregular polygons are also divided into convex and hand side line stand in lines which meet in a square also con. non-convex ; or, those whose angles are all salient, and those taining 24. The reason of this is that when the product of two of which one or more are re-entrant. The irregular polygon numbers is required, it is indifferent which we consider to be the (Fig. 20) has its angles at B, C, and D, salient; and its angles multiplier and which the m licand. Thus, 4 added to itself 6 at A and E, re-entrant. times, is the same as 6 added to itself 4 times. The truth of 34. Polygons are also divided into classes, according to the this may be seen, perhaps, more clearly as follows :number of their sides ; as, the pentagon (Fig. 21), having five If we make four vertical rows containing six dots each, as sides; the hexagon (Fig. 22), having six sides; the heptagon represented in the figure, it is quite evident that the having seven sides; the octagon having eight sides ; and so on. whole number of dots is equal either to the number According to this nomenclature, the triangle is called a trigon, of dots in a vertical row (6) repeated 4 times, or to and the quadrangle a tetragon. the number of dots in an horizontal row (4) repeated six times. And the same is clearly true of any other two numbers. LESSONS IN ARITHMETIC.-IV. Hence we talk of two numbers being multiplied together, it being indifferent which we consider to be the multiMULTIPLICATION. plier and which the multiplicand. 1. The repeated addition of a number or quantity to itself is 4. If several numbers be multiplied together, the result is called multiplication. Thus, the result of the number 5, for called the continued product of the numbers. Thus, 30 is the instance, added to itself 6 times, is said to be 5 multiplied by 6. continued product of 2, 3, and 5, because 2 x 3 x 5 = 30. 5 + 5 + 5 + 5 + 5 + 5 = 30, or 5 multiplied by 6 is 30. N.B. On learning the multiplication table, let the following facts be noticed : When the numbers to be multiplied are large, it is evident that the process of addition would be very laborious. The process adding a cipher to the number. The product of any number multiplied by 10 is obtained by of multiplication which we are going to explain is therefore, in reality, a short way of performing a series of additions. Let it, The first nine results of multiplying by 11 are found by merely The results of multiplying by 5 terminate alternately in 5 and 0. then, be borno in mind, that multiplication is, in fact, only. repeating the figure to be multiplied. Thus, 11 times 7 are 77. addition. 2. Definitions. The number to be repeated or multiplied is figure regularly decreases, and the left hand figure increases by In the first ten results of multiplying by 9 the right hand called the multiplicand. The number by which we multiply is 1; also, the sum of the digits is 9. Thus, 9 times 2 are 18, called the multiplier: it, in fact, indicates how many times the 9 times 3 are 27. multiplicand is to be repeated, or added to itself. The number produced by the operation is called the product. 5. It is evident that (as 2 X 3 X 5 = 30, and 2 X 3 = 6, and The multiplier and multiplicand are also called the factors of which the product another, 6, for instance, it will be the same thing if we multiply 6 X 5 = 30] in multiplying any number, 5, for instance, by is composed, because they make the product. Thus, since 5 multiplied by 6 is 30, 5 and 6 are called Thus, the product of any number multiplied by 28 might be got it successively by the factors of which the second is composed. factors of the number 30. The sign X placed between two numbers means that they by multiplying it first by 7, and then multiplying the result by 4. are to be multiplied together. 3. Before proceeding farther, the learner must make himself annexing a cipher to the number. The product of any number multiplied by 10 is obtained by familiar with the following table, which gives all products of therefore, multiplied by 100 will be obtained by adding two The product of any number, two numbers up to 12: ciphers, because 10 x 10 = 100; first multiplying by 10 adds MULTIPLICATION TABLE. one cipher, and then multiplying the result by 10 adds another cipher. Similarly a number is multiplied by any multiplier 5 10 which consists of figures followed by any number of ciphers, by first multiplying by the number which is expressed by the figures without the ciphers, and then annexing the ciphers to the result. Thus, 5 times 45 being 225, we know that 500 times 45 is 22500. 12 27 | 30 6. The process of multiplication which we now proceed to 4 8 20 24 28 explain, depends upon the self-evident fact that if the separate numbers of which a number is made up be multiplied by any 10 15 20 50 60 factor, and the separate products added together, the result is the same as that obtained by multiplying the number itself by 72 that factor. Thus 5 + 4 + 2 = 11 7 21 70 77 7 X 5 = 35, 7 X 4 = 28, 7 x 2 = 14, 35 + 28 + 14 = 77 = 7 x 11. 8 16 24 32 70 48 56 64 72 50 88 96 7. We shall take two cases : first, that in which the multiplier 18 27 36 45 63 72 81 | 90 99 108 consists only of one figure ; and, secondly, when it is composed of any number of figures. 20 4050 60 70 100 110 120 Case 1.-Required to multiply 2341 by 6. 2341 = 2 thousands + 3 hundreds + 4 tens + 1 unit. 55 77 110 | 121 | 132 Multiplying these parts separately, by 6, we get 6 units, 24 12 48 72 34 96 108 120 132 144 tens, 18 hundreds, and 12 thousands, which, written in figures and placed in lines for addition, areTo determine the product of any two numbers by the above table, fir' one of the numbers in the top line reading across the page, and then find the other in the line on the left hand which runs down the page. Follow the column down the page in which the first number stands, and the column across the page in which the second number stands. The number standing in Giving as the result 14046 the square where these two columns meet is the product of the The process may be effected more shortly, as follows, in one two numbers. line; the reason for the method will be sufficiently apparent Thus, to find the product of 4 multiplied by 6; 4 in the top rom the preceding explanation : 1 2 3 7 8 9 11 12 6 8 10 12 14 16 18 20 22 3 6 9 |