Aucun and nul should be put in the plural only before such (6.) In speaking of Charles the Fifth of Gormany, and of the Pope Sixtus the Fifth, the obsolete word quint (fifth) is words as are not used in the singular, or have in the singular a used: Charles-quint, Charles the Fifth. different acceptation. Sixte-quint, Sixtus the Fifth. (4.) CHAQUE is of both genders, and is used only in the singular. It always precedes the noun, and cannot be separated § 27.- NUMERAL NOUNS. from it by an adjective or by a preposition. It should never bo We shall, in order to render reference easier, place here some used without a noun :'observations on nouns and adverbs of number. Chaque âge a ses plaisirs, chaque Every age has its pleasures, overy état a ses charmes situation its charms. (1.) The numeral nouns in use with the French are :Unité, unit. Quarantaine, two score. (5.) MÊME, placed before the noun, has the sense of same in Conple, paire, couple, pair. Cinquantaine, fifty. English. Placed after the noun it means, generally, himself, Trio, trio, three. Soixantaine, sixty. herself, itself, or themselves. It may often be rendered by the Demi-douzaine, half-dozen. Quatre-vingtaine, eighty. word even. When même is an adjective it may take the form of Huitaine, sight days. Une centaine, a hundred. the plural, but does not vary on account of gender :Neuvaine, nine (nine days of prayer). Deux centaines, etc., two hundred. C'est la même vertu, it is the same virtue, C'est la vertu méme, it is virtue itself. Deux milliers, two thousand. Le peuple et les grands n'ont ni The people and the great have les mêmes vertus, ni les mêmes neither the same virtues nor the same Trentaine, thirty vices, vices. (2.) The termination aine signifies nearly, and when added to (6.) It is at times difficult to distinguish même an adjective words of number is equivalent to the English some, in cases from même an adverb, which is invariable. [See $ 97 (2,) (3.)] like the following : I have some twenty books, i.e., about twenty (7.) NuL is a stronger negative than aucun. It agrees in books. J'ai une vingtaine de livres. gender and number with the noun which it qualifies. Like aucun, when relating to the subject of the sentence, it requires $ 28.-FRACTIONAL NUMERALS. ne before the verb :Un quart, one quarter. Un cinquième, one fifth. Nul homme n'est heureux ; nulle No man is happy; nothing can Deux quarts, tuo quarters. Deux cinquièmes, tuo fifths. chose ne peut le rendre tel. render him so. Trois quarts, three quarters. Un sixième, etc., one sixth, etc. (8.) Nul is sometimes used alone, in the sense of no one : Deux tiers, two thirds. Un centième, one hundredth. Nul n'est content de sa fortune, No one is pleased with his fortune, La moitié, the half. Un millième, thousandth, ni mécontent de son esprit. nor displeased with his own wit, (1.) It will be seen that, with the exception of tiers, quart, and (9.) PLUSIEURS is, of course, always in the plural. It does moitié, these numbers take the form of the ordinal numerals. not vary its form :They may, therefore, take the form of the plural when neces. Il faut bien qu'il y ait plusieurs There must necessarily be several sary raisons d'envui, quand tout le reasons for ennui, when all agree in (2.) The word demi, when used adjectively and preceding the monde est d'accord pour bâiller. yawning. noun, is invariable ; but it is joined to the noun by a hyphen: (10.) QUEL takes the gender and number of the noun to Une demi-heure, f., half an hour, Une demi-aune, f., half an ell. which it relates. It is sometimes immediately followed by its (3.) When coming after the noun to denote an additional half, noun, from which it may be separated by one or several words :it agrees in gender with the noun : Quel tableau ravissant présentent What a delightful picture the Une heure et demie, one hour and a half. les campagnes ! country offers ! Une aune et demie, ono ell and a half. (11.) QUELCONQUE is always placed after the noun, and (4.) When used substantively, demi may take the form of the plural : varies only for the plural :Cette horloge sonne les heures This clock strikes the hours and Toutes les jouissances sont pré- All enjoyments are preceded by et les demies. the half-hours. cédées d'un travail quelconque, some sort of exertion. (12.) QUELQUE in the sense of some (a certain number), or § 29.-ORDINAL ADVERBS. whatever, agrees in number with the noun :(1.) Premièrement, first; in the Quartrièmement, fourthly. first place. There is merit without elevation, Il y a du mérite sans élévation, mais il n'y a point d'élévation sans but there is no elevation without some merit. (13.) Quelque having the sense of about or some or however, is (2.) These, like adverbs of manner, are formed by the addition invariable :of ment to the feminine form of the adjective. Quel âge avez-vous ? Vous avez How old are you? You look well, bon visage! Eh! quelque soixante Oh! some xty years, § 30.-INDEFINITE ADJECTIVES. (1.) The indefinite adjectives are used when anything is to be hommes, ils n'oseraient paraitre do not dare to appear enemies of Quelque méchants que soient les Horrever wicked mon may be, they represented or referred to in a general or indefinite manner. ennemis de la vertu. virtue. They are (14.) TEL makes in the feminine telle ; in the plural mascuAucun, not any, not one. Quel, what. Quelconque, whatever. line, tels ; in the plural feminine, telles. It agrees with the noun Même, same. Quelque, some. which it qualifies : Telles lettres, such letters, (15.) Tout, meaning every, is of course always in the singu. (2.) Aucun is generally followed by a noun, with which it lar, but varies for the feminine :must agree. It is followed by ne when it comes before a verb : Tout citoyen doit servir son Every citizen should serve his Aucun homme, no man. Aucune femme, no roman. pays; le soldat de son sang, le country; the soldier with his blood, Aucun chemin de fleurs ne con prêtre de son zele. the priest with his seal. No fiovery path leads to glory. (16.) Tout, in the sense of all, agrees in gender and number (3.) Aucun is by the French authors sometimes uscd in the with the noun to which it relates : Il était au-dessus de tous ces He was above all those rain objects Ils ne peuvent souffrir aucunem. They can bear no legitimate do. vains objets qui forment lous les which form all the desires and all pire légitime, ne mettent aucunes minion, set no bounds to their crimes. désirs et toutes les espérances des the hopes of men. bornes à leurs attentats. hommes, Deuxièmement, secondly . ans. duit à la gloire. plural : $31.-THE PRONOUN. n'ai pas de couteau sur moi. 15. Portez-vous un couteau sur vous ? 17. M. votre frère n'a-t-il pas (1.) The pronoun, in French, as in other languages, is a word 16. Je ne porte jamais de couteau. 18. Il n'a pas d'argent sur lui. 19. Voulez-vous used to represent the noun, in order to prevent its too frequent d'argent sur lui ? mettre ces crayons sur la table ou dessous ? 20. Je les mettrai dans repetition. le tiroir. 21. Combien avez-vous donné pour ce blé ? 22. Je l'ai (2.) The pronoun serves also to designate the parts which acheté à raison de vingt-cinq francs l'hectolitre. 23. La maison de each person or thing takes in speech. This part is called cette dame est-elle hors de la ville ? 24. Elle n'est pas hors de la person. ville, elle est dedans. 25. Votre sour n'a-t-elle pas mis les assiettes (3.) There are three persons: the first, or that which speaks ; sur la table ? 26. Elle a mis les assiettes sur la table et les cuillères the second, or that spoken to; the third, or that spoken of. dessous. 27. Avez-vous fait réparer votre maison ? 28. J'ai fait réparer (4.) There are five sorts of pronouns : le dedans, mais pas le dehors. 29. Combien vous coûte cette soie ? The personal; The demonstrative; 30. Je l'ai achetée à raison de cing francs le mètre. 31. Vous êtes. The possessive; The relative; vous marié à l'insu de votre père ? 32. Je me suis marié à son insu. The indefinite. 33. Avez-vous vendu mes livres à mon insu? 34. Je les ai vendus à votre insu. 35. Je les ai vendus à l'insu de ma seur. KEY TO EXERCISES IN LESSONS IN FRENCH. LESSONS IN MUSIC.-XXV. 1. Avez-vous acheté cette maison à l'insu de votre père? 2. Je l'ai THE DEAD MARCH IN “SAUL"-BOYCE'S CHANT. achetée à son insu. 3. Avez-vous oublié de fermer la porte de devant? The strikingly effective piece which follows, adapted from 4. J'ai fermé la porte de devant et la porte de derrière. 5. J'ai apporté Handel's “ Dead March " in his oratorio of “Saul,” and tous mes livres, excepté deux ou trois. 6. Votre frère occupe-t-il le reprinted from Mr. Curwen’s “ People's Service of Song,” will devant de votre maison ? 7. Il occupe le derrière. 8. Qui avez-vous rencontré derrière cette maison ? 9. Je n'ai rencontré personne derrière exhibit to the pupil the effects of transition in a still more la maison ? 10. Ce monsieur demeure-t-il derrière votre maison ? striking manner. (The key-note F is in the lowest space, except 11. Personne ne demeure derrière notre maison. 12. Il n'y a pas de in the bass. There it is in the fourth line from the bottom.} maison derrière la vôtre. 13. Avez-vous un couteau sur vous ? 14. Je Boyce's Chant illustrates both transitions. M. 50. EXERCISE 45.-DEAD MARCH IN SAUL (IN THE NEW NOTATION). KEY F MAJOR. [MUSIC FROM HANDEL. WORDS BY KELLY.] To the following music may be chanted—Psalms li., cxxxix. ; Isaiah lxii. 1—7, 9—16 ; Romans xi. 33–36; 1 Corinthians xv. 51-58; 1 Thessalonians iv. 13—18. с D M P H E AB. B В D EXERCISES IN EUCLID.-IV. to on. Therefore in the two triangles O AL, O AM, sizce o L, L A are equal to OM, MA, each to each, and oa is common, PROPOSITION XXI.-Given two straight lines A B, AC (Fig. therefore the triangles are equal (Euc. I. 8); therefore angle 20) meeting in A, and another line D Е, of limited length. Required OAL= angle O AM. Q. E. D. to describe an isosceles triangle A L M, such that a L may coincide Note.-It will be observed in the figure that on coincides with A B, and AM with AC, and L M may be equal to D E. with o a, but this is not assumed in the proof. This proposition Take two points P, Q in A B, AC, such that AP=AQ. Join will be proved in a future article for any triangle not isosceles. PQ, and produce po to R, so that PROPOSITION XXV. - In the figure of Euc. I. 1, if the circles RM cut Ac in M; then if Al be radii of equal circles. Hence 8 taken in A B equal to AM, ALM ABC, ABF are equilateral tri. will be the triangle required. Fig. 20, angles, and therefore also equi. For because the angles at the angular (Euc. I. 5, Cor.); but, base of an isosceles triangle are equal (Euc. I. 5), and that AQP by Euc. I. 32, the angles of & =RQM (Euc. I. 15), and that APQ=QRM by construction, triangle are together equal to therefore angle MRQ = MQR; therefore, by Euc. I. 6, MR = two right angles; therefore MQ. But since AM = A L, and A Q = A P, therefore, by Axiom each angle of an equiangular III., MQ=PL; therefore also RM=PL (Euc. I. 28). Also, since triangle is equal to one-third Fig. 24, angle A P Q = Q R M, therefore a L is parallel to R M (Euc. I. of two right angles--i.e., each of the angles CA B, B A F is equal 27); therefore Rm is equal and parallel to P L. Join pm, then to one-third of two right angles. Therefore angle CAF equals the angle RMP= angle MPL (Euo. I. 29); also side RM= two-thirds of two right angles; hence, by Euclid I. 13, angle side PL, and side m pis common, therefore base RP = base ML-FAH is equal to one-third of two right angles; therefore angles i.e., ML = D E (Euc. I. 8), and AM=AL; therefore A M L is A U F, AF1 are together equal to two-thirds of two right angles the triangle required. Q E. F. (Euc. I. 3), but they are equal, since AH=AF; therefore PROPOSITION XXII.-If A BC (Fig. 21) be an isosoceles triangle each of them is equal to one-third of a right angle, hence the of vertex A, with b c the base pro triangle AHF is equiangular, and therefore also eqnilateral duced to D, and if from centre c, at (Euc. I. 6, Cor.); therefore HF is equal to , or equal to distance c A, a circle be described Q. E. D. cutting BA produced in E, then PROPOSITION XXVI.-If in a triangle A B C (Fig. 25), B C be exterior angle EC D shall be equal bisected in G, and A G joined, and the angle BAG be equal to the to three times angle A BC or A C B. angle C AG, then B A shall be equal to CA. For if A B be not equal For since c A, C E are radii of the to A C, one of them must be greater. Let A B be the greater; from same circle, angle CA E = angle Fig. 21. A B cut off Ah equal to a C. Join C H, GH; then because An= CEA (Euc. I.5); but exterior angle A C, and ag is common, also included angle GAH= included angle CA E equals two interior and opposite angles ABC and A CB= GAC, therefore base GH=GC (Euc. twice angle A C B (Euc. I. 32); therefore angle CEA equals twice I. 4). But a C=G B, therefore G H = angle A B C Or A CB. But exterior angle E c d equals two interior GB, and angle G B= angle GBH(Eac. and opposite angles ABC and A EC (Euc. I. 32) equals three 1. 5). Also since G H=G C, angle GCH times angle A B C or ACB. Q. E. D. = angle guc (Euc. I. 5). Therefore PROPOSITION XXIII.-In the figure of Euc. I. 5, draw cl angles GIC and GHB are together (Fig. 22) at right angles to CB, meeting BA equal to GBH and GCH together- Fig. 25. ALC; therefore, taking away A CB, we have Note.—This kind of proof is called “ Reductio ad absurdum." Fig. 2. remainder ACL= = remainder A LC; therefore PROPOSITION XXVII.—Given two straight lines, A B, 40 (Fig. AC=AL (Euc. I. 6). Q. E. D. 26), meeting in A, and another straight line, D, of limited length. PROPOS ION XXIV.-If in the figure of Euclid I. 5 the Required to form a right-angled triangle, of which the base sball angles FBC, BCG (Fig. 23) be bisected by lines co, B 0, meet- coincide with AC; one side shall ing in o, then O a shall bisect the angle BAC. Draw O L, O M, coincide with A B, and the other side 5 on perpendicular to GC, C B, B F, then in the two triangles be equal to D. OLC, Onc, because the right angle oLc= = right angle ONC, From a draw A E at right angles and angle ico= angle nco, each being by construction half to AB (Euc. I. 11), and from a E cut the angle Lcn, also side off A F, equal to D (Euc. I. 3); from oc is common, therefore F draw FG at right angles to A E, triangle o LC= triangle cutting Ac in G (Euc. I. 11); from Onc (Euc. I. 26); there G draw G H perpendicular to AB (Euc. fore cl=CN, and OL= I. 12); then will A H G be the triangle on. Similarly, BM=BN, required. For since 6 HA, HAF, A F G are right angles, A IGF and om=ON; but since is a rectangular parallelogram; hence in the two triangles FAG, ON B, ONC are right angles, AGH, because the angle FAG is equal to alternate angle A GY and angles OC N, O BN are Fig. 23. (Euc. I. 29), and right angle AF G equal to right angle ANG equal, being halves of the also side Äg is common, therefore the triangles are equal in equal angles LCN, M B N, and side on is common (Eac. I. 5), every respect (Euc. I. 26). Therefore side A F=G H, but af therefore the triangles O NB, ON care equal; therefore cn=NB. =D; therefore Gu=D, and Al coincides with a B, and AG But Lo=cn, and mB=BN; therefore L'c=BM. But ac=1 with ac, and A nG is a right angle; hence A G is the triangle AB; therefore AL=AM, and o L, O M are equal, each being equal required. Q. E. F. H D G B IH A Fig. 26. N M B A B F PROPOSITION XXVIII.-If, in the figure of Enc. I. 1, AB in a door-post. It is by shells of this kind that the huge blocks prodaced cat the circles again in D, E (Fig. 27), and the circles of stone used in building the Plymouth Breakwater and some of cut again in F, the figure C EFD shall be a rhombus, having the new military works are slowly but surely being reduced to a each of the angles at c and F half the angles at D and E. species of stone honeycomb. Not only stone but solid and Join AF, B F, CF; then because the angles CA B, CAD are dense-grained timber is readily bored into by the pholades. We equal to two right angles, and similarly C B A and C B E (Euc. stated in our last paper on this subject that much difference of 1. 13), therefore angles C A B and CAD are equal to CBA and opinion and scientific argument had arisen on the subject of the CBE. But CAB=CBA (Euc. I. 1), therefore remainder CAD= boring powers of these curious creatures. Some philosophers remainder C B E; and since A B, AC, have stoutly maintained that the animal secreted a flaid of “acid' AD, BC, B E are all equal, being radii reaction, which possessed the power of so acting on the con. of equal circles, therefore in the two stituents of wood, stone, amber, wax, and gum resins that they triangles CAD, CB E, because sides became sufficiently soft and disintegrated as to admit of the EC A, A D=sides C B, B E, each to shell, together with the mollusk inhabiting it, passing freely into each, and included angle CAD and through the substance acted on. Others have maintained included angle C BE, therefore base that the minute rasp-like teeth, or asperities, with which shells CD = base C E (Euc. I. 4). By of this kind are armed, being constantly brought to bear on the Fig. 27. an exactly similar process of reason exposed surface of the stone at the bottom of the perforation, ing, DF=FE, and because CB= were alone the agents in force to deepen the tube. Another set BF, and B D is common, and included angles CBD, FBD are of investigators have stated that the borders of the soft coating equal, each of them being the angle of an equilateral triangle, and or mouth of the mollusk, aided by its short, stout foot, were therefore equal to one-third of two right angles (Euc. I. 32), the means employed. It has been also urged that the constant therefore base c d equals base D F, similarly CE=EF; there- and decomposing action of minute currents of sea-water passing fore CD, DF, FE, EC are equal, and CDFE is a rhombus. through the siphon-like tissues of the animal brought about the Again, because a C=AD, the angles A CD, A DC are equal (Euc. fretting action requisite to form a hole. 1.5); but angle C A B is equal to the two interior and opposite We have broken the pholas shells from stones of a sharp, angles A C D and ADC (Euc. I. 32), therefore the angle CAB is sand grit, which would be found to grind the hardest steel double either of the angles A C D Or ADC. Similarly the angle rapidly away. These shells we have examined under a powerful CBA is double either of the angles B C E OT BEC; therefore the lens. The asperities on them have been by us most carefully angle AC B, which is equal to either of the angles C AB Or A B C, scrutinised, but without our being enabled to detect the slightest is double either AC D Or ADC, Or B C E Or B EC; hence Ac B is evidence of. wear and tear by friction. Every minute point equal to the sum of A C D and BCE, and is also equal to the remained as sharp as a new needle, and bore no traces of having sum of ADC and BEC; therefore the whole angle DCE is cut through a mass of stone thick enough to have destroyed the double A CB, and is therefore four times the angle C D E or points of a dozen engraving tools. Then when two of these CED; similarly, the angle D FE is double A F B, and is therefore shell-coated miners so drive their galleries that they intersect four times the angle F D E or F E D. But FDE or F E D is equal each other, the more powerful workman of the two, ignoring the to C D E Or C E D, hence either of the angles DCE or D FE is presence of his weaker fellow-labourer, works on, bores forwards, double either of the angles C D F or CEF. Q. E. D. and not only tunnels the rock, but the shells and soft tissues of Our next ačticle will extend as far as Euc. I. 40, and we shall his neighbour, literally boring him through and through. We give solutiong of the following propositions : have never been able to detect by the ordinary tests any “acid” PROPOSITION XXIX.—To trisect a given right angle, that is, in the water thrown off from the siphon of the pholas. to divide it into three equal parts. The rock-boring snails (Helix saxicava) before described by PROPOSITION XXX.-If two right-angled triangles have one us, although forming deep tubular chambers in hard, dense rook, side and the base in one equal to one side and the base in the have no currents of sea-water to aid them, neither have they other, each to each, they shall be equal in every respect the same rasp-like and rounded character of shell. The ring PROPOSITION XXXI.-The straight lines which bisect the like mouth or portal of a snail-shell could but grind and wear angles of a triangle meet in a point. down (supposing the file process to be that in force) in an PROPOSITION XXXII.—The straight lines drawn perpen- uneven circle corresponding with the shell border. dicular to the sides of a triangle through their middle points To illustrate our point a little more clearly, let us place a commeet in a point. mon wine-glass or a metal thimble mouth downwards on a piece PROPOSITION XXXIII.-The straight lines which bisect one of soft Bath-brick, and then proceed to work it round until it interior and two exterior angles of a triangle meet in a point. penetrates the substance on which it is placed. On examina PROPOSITION XXXIV.-If two triangles have one side, and tion of our work, we shall find a groove corresponding to the one angle in the one equal to one side and one angle in the edge of the circle of friction, and an even, table-like middle no other, and likewise their areas equal, then shall also their other deeper than the plane surface of the brick, which, as the groove sides and angles be equal each to each. deepened, would enter the mouth of the snail-shell, press up the PROPOSITION XXXV.-If the bases of two equal triangles be inhabitant, and finally stop his boring operations altogether. in the same straight line, and the line joining their vertices be. We find, however, on examining a real snail-tube, that it is, parallel to this line, their bases will be equal. although high and dry on land, formed much like that made by PROPOSITION XXXVI.-In the figure of Euc. I. 5, if ac be the pholas, and the bottom of the excavation, instead of being bisected in , and co be equal to c A, then BG shall be equal even, is cup-like in form, the centre, where no ring-shaped shell could touch, being the deepest point; and, curiously enough, the tracks made by the snails in going to and from their winter RECREATIVE NATURAL HISTORY. retreats, year after year, are of a distinctly grooved form. Two snails are not uncommonly found in the same tube, but, unlike SOME LAND, SEA AND FRESHWATER SHELLS, WORMS, the pholades, they never in any way interfere with each other. AND TUBE-DWELLERS (continued). It has been stated that a distinct acid reaction has, by the aid of Few of our readers who have investigated the habits of the litmus paper, been detected in the fluids given off by the boring deeply-interesting and curious creatures found amongst the rocks snail. To this statement we attach bat little importance. Visit and rock pools of our own coast, will have failed to notice the a nest of black wood-ants, place a piece of slate in a split deep tubular excavations made in the rock by those accomplished stick, hold it over the ant-hill, and then irritate the community; and industrious borers, the Pholas family. In different localities acid enough to act strongly on litmus paper will be at once dewe find two of these stone perforators (Pholas dactylus and Saxi-posited on the stone. And yet ants do not bore holes in rocks. cava rugosa). The former of these we find prosecuting his The composition of the secretions of living organisms is, labours both amongst the chalk rocks and red sandstone of the in many instances, entirely beyond the powers of the most southern coast of England, whilst the latter, not content with accomplished chemist either to imitate or correctly lay down. attacking substances of an ordinary degree of hardness, proceeds The silkworm spins for itself a cocoon, or capsule, in which to to operate on the compact, hard limestone rock, cutting his way rest until the period for change into the moth stage arrives. deeply into it, just as a skilful carpenter bores an augur-hole Examine one of these cocoons, and reflect as to how a tiny, to twice BH. |