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TION OF A FIG.
observations mado on young dragon-trees, the growth of which gains little knowledge from this remark beyond the knowledge is remarkably slow. What grand, what stupendous thoughts of an, at present, unmeaning name; and as we do not intend does & contemplation of this fact awaken! When did this that any names in this series of papers on the Science of Botany monster first begin to grow? How many thousand years have shall be unmeaning, we will at once proceed to explain what a rolled over its weather-beaten head? We are afraid to specu- torus is. late on these points, but will content ourselves by saying that, Torus, then, is the Latin word for bed, and signifies that poraccording to the most reasonable evidence which can be adduced, tion of certain flowers upon which the flower itself reposes or this great dragon-tree began to grow long, very long, before the grows. Take, for example, the marigold, and strip off all its creation of man. Yet this monster is a lily!
floral parts; there will then remain underneath a flat, fleshy The student will admit that, supposing our previous remarks expansion, called the torus. In the case of the marigold the to be correct, our ordinary notions concerning the similarities or torus is flat; but the reader may easily conceive that it might dissimilarities of vegetables—in other words, their alliances, and have been round or approaching to rotundity. In the marigold it as a consequence their classification are very incorrect. Not is leathery and nauseous, but the reader less incorrect are some of our common ideas regarding the will as easily conceive that it might have similarities and dissimilarities, or the alliances, of the parts of been fleshy and delicious, as indeed we which vegetables are composed. For example, do we not com- find it to be in the strawberry. Analysed monly speak of onions and potatoes as roots ? Yet they are not thus, we find a similarity between the roots, nor are they similar, far less identical, in character. The strawberry and the marigold that the onion is a bulb, or underground bud, and the potato is a tuber non-botanical reader would have little or knotty excrescence developed underground, from which the suspected. Nor is the similarity forced; roots and stems of the potato plant respectively spring. Why it is natural, and loses nothing by the are they not roots ? the learner may ask. The reason why will fullest investigation which the learner appear by-and-by: to explain these reasons is an object, and can devote to it. Thus, we dare say, the one of the main objects, of botany. We merely cite the ex. reader has watched the progress of a ample now for the purpose of making known in a striking marigold to maturity; has noticed the manner the incorrectness of many notions we are in the habit flowers blown away, one by one, and of entertaining.
nothing but the stem, the torus, and the Again, do we not in ordinary language term the strawberry little seed-like things embedded upon the
2. LONGITUDINAL SEC and the fig fruits ? Yet neither is a fruit.
torus remain. These little things, like "Not a fruit!” the learner exclaims," do we not eat them ?” the hard excrescences on the strawberry, Well, surely, our reader would not limit the term fruit to some look so much like seeds, that they might thing which grows on a vegetable, and which is good to eat. We be taken for such. However, we are never to assume because think he will admit that the bunches of apples, as they are a thing is small that it is imperfect. If these so-called called, which grow on potato stems, are the fruits of the potato seeds be dissected and examined, they will be found to be real platt; yet potato apples are not good to eat. He will admit fruits, as much as the apple or the pear, and so contain seeds that the banches which grow on ash-trees are the fruits of those internally. treen, yet they are not good to eat. Finally, not to multiply And now for our other example, the fig. What is the fig? exanples unnecessarily, he will admit that acorns are the fruits Not a fruit certainly, although the freak of Nature here, if we of the oak-tree ; and although our ancestors, the ancient Britons, may without disrespect use such a term, is different from those are known to have eaten them, yet all we can say upon that which have come under our notice hitherto. Let us cut open a point is, that one pities the bad taste or the hard fortune, as the fig; what then do we see? Why, little things very similar in case nay be, of our forefathers.
appearance to flowers, at the base of each of which there is a If strawberries, then, and figs are not fruits, what are they? hard nut-like thing which cracks between the teeth. Flowers Why, the fig is to all intents and purposes a compound flower, indeed they are, and the nut-like things are fruits, the edible as much as the dandelion is a compound flower; and a straw. portion of the fig being a torus; so that if we assume the berry is something like a fig turned inside out; but the learner strawberry to have had a flat torus instead of a knob-like one, shall judge for himself.
and that the flat torus had been turned outside in, in such a The strawberry plant bears, as we all know, a very evident, manner as to form a bottle with a very narrow mouth, we should a very pretty flower, the petals or flower-leaves of which drop- have had a result very much resembling a fig in structure and ping of, we ultimately get something which is good to eat, and general appearance. which we term the strawberry fruit.
Even the delicious pine-apple can hardly be termed a fruit. Why, then, is it not a fruit ? We will see. If it be a fruit, Each pine-apple certainly contains many fruits, one correspondit shoud contain seeds; but on cutting it open we cannot find ing with each lozenge-like marking; but the main bulk of the any. Iere, then, the learner would be puzzled if botany did pine-apple, that which we find so delicious to eat, is only an not cone to his aid. General principles have to be appealed to, assemblage of juicy fructs, as botanists call them, the exact and the appeal will not be made in vain,
counterpart of those little scales which, when tightly compressed Whilst conjecturing within ourselves the botanical nature of together, form the cup of the acorn.
the strawberry, and trying to find out We are sure, then, that sufficient has been stated to make ap-
LESSONS IN GERMAN.-I.
its seeds externally; and amazingly English languages. The learner will be supplied, throughout the 1. TOLUS OF THE MARIGOLD. like seeds do these little protuberances various exercises, with the materials necessary for their due per
appear. They are not seeds, never- formance. Every section is headed with the statement and theless: they are fruits, the real strawberry fruits ; but so little illustration of all new principles involved, with an explanation adapted or eating are they that the lover of strawberries wishes of words and phrases, and a vocabulary alphabetically arranged. them yew far away. Then what is the edible portion of the To render these lessons complete, there will be given at the strawbery ? Botany answers this question satisfactorily, and end a series of reading lessons, each accompanied by a full makes al clear. It is the juicy torus of the plant. The reader vocabulary. The whole is specially intended for thoso
who aim at the acquisition of the German language without a are produced by a union of e with a, 0, u (also au) respectively, teacher.
and the e is expressed by two dots; thus, å, ö, ü (and äu). The SECTION I.
capitals Ú, Ó, ů, are not much in use, and the student should
never make use of them in writing. GERMAN ALPHABET. German. English, Pronunciation. Examples.
12. Ae, å, as a in top, tack, carry. Ex., Aerger, vexation; Fähre, ah Alt.
ferry. b b bay
13. De, ö, as u in return. Ex., Del, oil ; Pöbel, populace : tödten, tsay eder.
to kill ; Röhre, pipe; Köhler, collier. d day
14. Ue, ü, has the sound of the French u in vu, tribu, élu. Ex., ey (as in prey) Gben.
Uebung, practice; müde, weary; führen, to guide. f f eff Fett.
Sounds of the Consonants. gay
15. B, 6; D, 0; F, f; K, #; 2,1; M, m; N, n; Þ, ; 0,9; X, h hah
r, are sounded as b, d, f, k, l, m, n, p, q, a, in English. i i (as in pique) Ihnen.
16. O, c, before a consonant, at the end of a syllable, or before j yote
a, o, u in the same syllable, sounds like our corresponding k kah
letter in like position. Otherwise it sounds like ts. Ex., 1 ell
Ceder, cedar; Cigarre, cigar; Cymbal, cymbal; special, M
17. O, g, sounds like our g in gild, foggy, etc., but never as in oh Dhr.
gem, ginger, etc. When preceded by n in the same P
syllable, it sounds like our g hard in like position: as in q koo
Angst, anxiety; singen, to sing ; bringen, to bring; Ningel, err (as in error) Grretten.
ringlet, etc. When g, in the middle or at the end of a s 8 (see 21. S) s
syllable, is preceded by any letter except n, its sound t tay
Thee. (see 18. 5.)
approaches that of the Greek x (pronounced ki), or the u
o (as in do)
still more guttural ch (see 26. ch.): Tag, regnen, Magd, V fow (ow as in mow) Volk,
Jagd, möglich, etc. The learner should avoid confounding W
the pronunciation of Magd, Jagd, etc., with that of Macht,
18. H, h, in the midst and at the end of a syllable is silent, but 3
serves to lengthen the preceding vowel. Ex., lehren, to In German every letter, with the exception sometimes of e teach ; ohne, without; Thee, tea. and ḥ, is pronounced. (See 3. i, 9. ie, and 18. h.)
19. 3, i, sounds like y consonant. Ex., Jahr, year; Januar, The printed capitals of i and i, in German, are in form alike.
January ; jung, young. DIPHTHONGS.
UMLAUTS (12. Ae, etc.) 20. R, r, is uttered with a trill or vibration of the tongte, and ai, au, ei, eu, iu.
å, o, ů.
with greater stress than our r. Ex., Rohr, reed; Rath,
council ; reif, ripe. COMPOUND CONSONANTS.
21. S, s, at the beginning of a syllable followed by a vowel, 193 o, ich, 1. ftB. $.
à sound between that of 2 and s. Ex., Sohn, son; sieben, ck, sch, ss, st, sz, tz.
seven: otherwise it sounds like s; as in Gas, gas, Strom, SECTION II.
stream. Note that at the end of a syllable s is substi.
tuted for s; as above, Gas, etc.
22. I, t, sounds like t in tent. Ex., Tert, text. In the position Sounds of the Vowels.
where in English t sounds like sh, t has the sound of ts.
Ex., Station, station; Nation, nation. 1. A, a = a, as in far, father. Ex., Markt
, market; Aal, eel; 23. v, w, sounds like f, as in five. Ex., Vater, father ; vergeben, Bahn, road; Blatt, leaf ; Abend, evening.
to forgive. It is only in words from the Latin and 2. E, e = e, as met, ferry. Ex., leben, to live ; Meer, sea ; Ehre,
French that y has a sound like that of the German w honour; besser, better ; Messer, knife. 3. 3, i = i, as in pique, pin. Ex., mir, to me; mit, with; ihn,
(see 24. W), as in Venus, Venus; Versailles, Tersailles,
etc. him; wider, against; bitter, bitter.
24. W, w, has a sound between that of our w and v. Ex., Welt, 4. 0, 0 = 0, as in no, door. Ex., Dfen, stove; Moos, moss;
world; Wasser, water, etc. Koble, coal; Port, port; Post, post-office.
25. 3, 3, sounds like ts. Ex., Salz, salt; Zahn, tooth; Zunge, 5. U, u = 00 or 0, as in poor, do. Ex., Blut, blood; Du, thou ; Uhr, watch; Hut, hat; gut, good.
tongue; zehn, ten. 6. 9, y = i (mostly in words from the Greek). Ex., Vsop,
Sounds of the Compound Consonants. hyssop; Styr, Styx; Ypern, Ypres.
26. Ch, ch, in primitive words, when followed by s, !, has the The sound of a vowel when doubled, is thereby lengthened ; sound of k. Ex., Dachs, badger ; Ochs, or Ochis ox. But as Aal, Meer, Moos ; followed by a double consonant, the vowels if s, &, be added by derivation, combination, o inflection, are usually shortened, as Blatt, Prett, Sinn, Gott, etc. See, how. dy has its guttural sound; as in hoch, nach, Nacht, Buch,
etc. Ex., Nachschrift (from nach, after, and Schrst, writing); Dissyllables (see vocabulary), unless otherwise noted, are
nacisinnen (from nach and sinnen, to think), etc. In words accented on the first; as leben, Ghre, etc.
from the Greek and French, ch retains its original sound; Sounds of the Diphthongs.
as in Charakter, character ; Charlatan, charlatan
27. Sc. ich, sounds like sh. Ex., Schug, shoe; Schiff, ship; 7. Ai, ai (sometimes aj or ab) = as nearly as in aye. Ex., Kaiser,
schon, already; Schule, school. emperor; Baiern, Bavaria ; Mai, May. 8. Au, au = ou, as in our. Ex., Haus, house Maus, mouse;
28. f (though compounded of sand 3) sounds like fl. ind is used
only at the end of a syllable. Ex., Maß, messure ; Fluß, laut, loud ; Faust, fist; Braut, bride.
river, etc. 9. Ei, ey=i or ei, as in fine, eider. Ex., Stein, stone ; dein, thy. 29. * (though compounded of t and 3) sounds like 3, but, like, ki
(ie = ie, as in pier, never as in pie. Ex., viel, etc.) 10. Gu, eu =
is only employed at the end of a syllable. Ex., Schuß, nearly to oi or oy, as in boil, boy. Ex., Beute,
Plaß, etc. Note that this letter being a couble conbooty; Leute, people; heuen, to hay. 11. Aeu, du : nearly to cu. Ex., Aeußerst, extreme; haufen, to
sonant, the preceding vowel is thereby shortened. hoard; Käufer buyer ; Häusler, cottager.
To aid in producing the sound of c), take for experiment the Sounds of the Umlauts (Umlaute).
above word Hoch: pronounce ho precisely like our word ho;
observing to give as full and distinct a breathing & the h at Umlaut signifies changed or modified sound. The Umlauts the close as at the beginning ; thus b-ob =
SOUNDS OF THE GERMAN LETTERS.
ever, 18. 5.
hoch. Except when
preceded by a, o, or u, as will be perceived by experiment, a honest voice to the closing strain. In public worship, too, no sligat hissing sound of s or sth naturally attaches to the ch, as in frowns or dissuasives could hinder him from “ doing his best” rett, reich, ict, Gricche, etc.
to join in the praises of God. He often wondered how it was EXERCISE 1.
that he came to be born with “no voice," especially when he altar, altar ; Paar, pair ; Ahle, awl; Balsam, balsam; baden, observed that the infants of the present day are so much more to bathe; Psalm, psalm.
highly endowed, every one of them who attends an infant-school Geer, host; mehr, more; edel, noble; Ende, end; Letter, letter; apparently taking for granted that he has a voice," and using Herbit, autumn.
it accordingly. As a religious man, also, he could not help Trinten, to drink; finden, to find; Biber, beaver; hier, here; noticing that one whole book of the Scriptures was written for Kind, child.
the promotion of public vocal praise, and that it abounded in Beot, boat; Hohl, hollow; oft, often; Hobel, plane; Roller, such expressions as this : "Let the people praise thee, O God: collar ; Koffer, trunk.
let ALL the people praise thee.” The example of Christ and the Fuß, foot; gut, good; unten, below; Pudel, poodle ; suđu, precepts of his apostles seemed also to set forth the same duty. cuckoo; Muth, courage.
“It cannot be," he sometimes reflected, " that the Father of all Nymphe, nymph; Rhythmus, rhythm; Sylbe, syllable; synonym, should command us to sing,' in addition to making melody synonym; Syrup, syrup.
with our hearts,' and yet give to so many of his children no si, ei. Main, Maine; mein, my; Laib, loaf; leib, body; Sain, Cain; voice !” Such thoughts as these led him to the conclusion that fein, no.
it is no practice” and “no cultivation,” rather than "no Bauen, to build; Mauer, wall; grau, grey; Raum, room; voice" and "no ear,” with which the majority of men are rauh, rough.
afflicted. In consequence of this, to the no small amusement of 11, eu. Räumig, roomy; reuen, to rue; Häute, skins; heute, to-day; his musical acquaintance, our friend was soon found to have Häuptling, chieftain.
become an attentive and painstaking member of a singing class. i, e. Nehre, ear (of corn); Männer, men ; leben, to live; Krähe, He was soon deep in “thirds" and "fifths” and “sevenths," crow ; nämlich, namely; nehmen, to take.
toiling at a series of the most unmusical exercises that could i. Löffel, spoon ; Deffnung, opening; öfters, oftener; röthlich, well be invented. But hope sweetens toil, and the expectation reddish.
of conquering at last gave to our friend courage and long Ulebel, evil ; fünf, five; Rüssel, proboscis ; Krüppel, cripple ; patience. When sixty laborious lessons, relieved by an occaJünger, disciple.
sional song, were over, he made the discovery that he had learnt EXERCISE 2.
a system,” that he had gained also some confidence and much Classe, class ; Greatur, creature; Criminal, criminal ; Lection, command of the organs of voice. But what did he know of lesson ; Calcutta, Calcutta ; Contract
, contract; Gur, cure ; of c), unseen before, and sing it? Alas! no. His labour had
music? Could he take the plainest psalm tune (not in the key Gement, cement; Cider, cider ; Cylinder, cylinder. Gabe, gift; gehen, to go; Giraffe, giraffe; geben, to give; not been lost, but it had produced small fruit. He could follow Gold, gold; groß, large ; Ring, ring ; bringen, to bring; without him. There was still an indecision and uncertainty
the “leader" more promptly and easily, but he could not go grün, green; grau, grey ; ruhig, quiet ; ewig, eternal; Berg, about his endeavours. He could seldom be sure whether he
mountain. V. Hase, hare ; hart, hard; Hunger, hunger; Horizont, horizon; not a few tune-books, which he had purchased in his hopeful
was right or not by half a tone. And many a choice song, and Mebl, flour ; mehr, more. 3. Jüngling, youth ; Jude, Jew; ja, yos; Joseph, Joseph; Juli, days, lay on his table unenjoyed because of this musical uncerJuly; Jurist, jurist.
tainty in which he was left. Once more, however, our friend R. Reif, ripe ; reich, rich ; Rest, rest; rar, rare; Rūdsicht, regard; effort which we shall prescribe; we, on our part, undertaking
has “taken heart," and has promised to follow the course of Form, form ; Räthsel, enigma. Š. Sattel, saddle ; Segel, sail ; Speer, spear; Sprofi, sprout; himself
, and to make good use of the books on his table. We
that he shall in that case be enabled to sing at first sight by starf, strong; Strumpf
, stocking; Süd, south; Neis, rice ; shall begin at the beginning, however, for your sake, gentle Straße, street; wissen, to know. Tisch, table; Tarif, tariff ; Tempel, temple ; Truppe, troop; reader, if you will join him in his efforts. We have no “ royal Titel, title; Devastation, devastation.
road" to music. No worthy attainment is won withont labour. B. Bampyr, vampire; Base , vase ; Vers, verse; Violine, violin; advantage when the common road is very circuitous, and
But we have a straight and clear road, and that is a great Visite, visit; Valuation, valuation. B. Wert, word; Wurm, worm; Wunder, wonder ; Wille, will; to ask of you: the first, that you will be content to learn one thing
abounding with needless hindrances. We have only two things Wagen, wagon ; Wanderer, wanderer. 3. Zinf, zinc; Zahl, number; zahm, tame; Zeit, time ; Zentner, at a time, instead of being impatient for knowledge not at the hundred-weight; Holz, wood.
moment helpful—perhaps, just then, only confusing to you; the Ch. Flachs, flax; sprechen, to speak; wachsam, watchful; Chor, choir; second, that when something is set before you to be done, you Chaussee, turnpike-road.
will really do it, instead of supposing it to be done, and going $. Somaft, shaft; Schatten, shadow ; Schnee, snow; frisch, fresh ;
on; for only “by doing we truly understand.” Scild, shield, sign. $.fi. Fleiß, diligence; Fließ, fleece ; lassen, to let; haffen, to hate ; Haß, hatred ; hablich, agly.
You must allow us to lay before you certain fundamental 13 Hiße, heat; Klog, log'; kişeln, to tickle ; schwaßen, to prattle ; method of teaching or of writing it,principles which would be
principles of music itself-of music considered apart from any schwißen, to perspire; kurz, short; schwarz, black.
true of music if Guido had never invented the “staff," and if 3 4 15 ( [.
“ crotchet” and “quaver," "flat” and “sharp,” had never Vier Jahre bleibt er aus,
Erst weiß wie Schnee,
been heard of. Dann kommt er nach Haus, Dann grün wie Klee,
You know what the difference between “high" and "low" Und zeigt sich wieter,
Dann roth wic Blut,
in music. The “squeak” is high, the “growl” is low. RecogIm Kreise seiner Brüder.
Dann schmeđt es gut.
nise this difference to yourself now by singing first a high and then a low note. Between the highest and the lowest sounds
which the human ear can appreciate, an indefinite number of LESSONS IN MUSIC.-I.
other sounds may be produced. But how, out of this vast chaos
of possible sounds, are the distinct and choice notes of a TUNE to We have a friend, who was long persuaded by his relatives, start into life and power? The question is thus answered. who were all “musical,” that he had “no voice." Any innocent Before a TUNE can be created, a certain sound, whether high or attempt of his to unite in the vocal pleasures of the
family low in pitch, must be chosen and fixed as the KEY-NOTE (somecircle was instantly checked by some compassionate expression times called the governing note, and in books of science the or imploring look. He humbly acquiesced in this judgment of tonic) of the coming tune. Immediately, according to those laws his friends, but found it often difficult to resist the sympathy of nature by which God has tuned our ears and souls, six other of song, and sometimes startled the singers by adding his notes spring forth, at measured distances from the key-note,
FIRST PRINCIPLES OF MUSIC.
claiming the sole right of attendance upon it. Let this be For the present, we wish your attention confined to the three clearly understood. Any sound may be taken for the KEY notes, Don, ME, and Son, the first, the third, and the fifth. NOTE ; and that being fixed, the places of the six other notes They are the strong notes of the scale, on which, as you will are known.
afterwards learn, the others lean. We may call them "the The common human ear throughout the world is pleased when framework of the building.” When sounded together they are these sounds attend that key-note, and is displeased when other commonly called the “chord of the tonic," tonic being the sounds, not holding the same relation to the key-note, and not scientific name for key-note. Chiefly by these notes your voice standing at precisely the same relative pitch, are used in their must be tuned. Take, then, some low sound of your voice for stead; for even an uncultivated ear would promptly mark the the key-note, or Don, and try to sing the following exercises, difference between the accurate singer and the inaccurate, pointing to the notes on the scale given above, as you sing. This between the singer in tune and the singer out of tune.
pointing on the scale is more important than you would at first This distinct arrangement of six sounds around a key-note is imagine. In no other way can you obtain so clear a notion of called the musical “scale.” It may be high in pitch in one the relative position of notes. If previously uninstructed, you tune, and low in another, but the relative position of its notes must ask some musical friend to sing these notes to you, or remains unchanged. These notes may be reproduced in replicates play them on an instrument for a pattern. Do not, on any or“ ootaves" of higher or lower pitch, but they still retain the account, however, sing with him or let him sing with you. same relation. Transition or “modulation" (which will be Remember that you are learning to sing alone. Your friend afterwards explained) may change the key-note in the course of will know what notes to play when you tell him D, F sharp, a tune, but the new key-note governs its dependents exactly as A, and upper D'; or, if he prefers it, C, E, G, and upper C!. the old one did. Every apparent exception only proves the rule. You will notice that when a note is repeated in a higher pitch, This one scale is the foundation of all music. Some speak of this we put this mark (1) above it: thus, Dow! You need not scale as though it were of human invention ; but if so, how is trouble yourself with the "staff” of five lines at present, except it that every newly-discovered nation is found either using it to notice that Dou is printed as a square note. (if they are musical at all), or possessed of ears which readily
EXERCISE 1. approve it? How is it that the Chinese or Indians have not “invented” some other scale ? The truth is, some of these nations do omit a note or two, but they do not alter the rest; and when the question is fairly examined, it is found the omissions were caused by their rude and incomplete instruments, rather than by defective ears. Again, let me ask, going back
Don to the time of the ancient Greeks, of whose musical notation there is not a remnant from which we could have copied, how is it that we learn, from their philosophical treatises, that the scale which the people used was the same as ours ? Could not that refined people have "invented” something better? Are we not
Don right, then, in calling it the scale of all nations and of all times,
Note.—Sing these notes first slowly, then quickly, and again the scale to which the ear and soul of man are tuned by the with a sound “long drawn out.” Do not be disappointed if all-wise Creator ?
your friend pronounces you inaccurate in the first and second When we examine its structure more closely, we find other notes, though they are the easiest. Let him patiently set the proofs that it comes from the hand of God. Like many of “pattern" of those two notes again, and, if need be, many his works—the rainbow, for instance—it seems to the careless times again. Master one note at a time. Some pupils require observer irregular, but discloses a beautiful harmony and pur several lessons, with much patient “patterning” of the teacher, pose to him who is more thoughtful. The distances in pitch and much careful listening, followed by vocal effort of the (that is, highness or lowness of sound), or, in other words, the learner, before this exercise is perfectly done. intervals between the notes of this scale, are very delicately arranged. In another lesson we shall be able to describe its
EXERCISE 2. structure more minutely; but let it suffice for the present to say, that the simplest measurement of the scale in plain figures
is that which divides it into fifty-three degrees.
Such a division is only inaccurate to the extent DOH
Dон ME SOH ME SoH ME Doł? of being about one-third of a degree too large. TE If you will make use of the sol-fa syllables to
represent the notes of this scale, Dou standing for 9
THE KEY-NOTE OF A TUNE, at whatever pitch it is LAH taken, then the number of such degrees between
Doh Son ME Son Son ME Дон each couple of notes may be set forth by the figures at the side. Why the scale of music found
Note.—You observe the upright bars. Sing the note immeSOH most acceptable to human ears should be thus diately after them with a stronger accent or force of voice than
curiously and delicately formed, and why it does the others. You notice that two of the notes on the " staff” of
not exhibit a greater apparent uniformity, we five lines are open, and that the names beneath are followed by FAH cannot tell. It is an “ultimate fact” of philo- a stroke of "continuance." Sing those notes twice as long as ME
sophy, like the structure of the rainbow. We must the rest.
these seven notes; and in a large work written
by Mr. Hay, of Edinburgh, a clear relationship DOH has been established between the principles of
Don Don ME ME Son Son ME beauty in the human form, and certain angles
founded on the proportions of the musical scale. Doubtless there are in the various departments of Nature certain uniting principles, certain secret affinities of things, which shall prove them all to have sprung from one creating Hand.
Doh Doh ME ME Sou Son Don It may, however, be noticed here, that every note of the scale
EXERCISE 4. sounds pleasantly, when heard at the same time with the keynote, excepting only Ray and TE; and of these, the most diff. Sing all these exercises again, while some one else repeats cult notes of the scale, more will be said when our lessons are the note Don for every note you sing. This we call “ tolling further advanced.
LESSONS IN GEOMETRY.-I.
become a “household word.” Scarcely less celebrated was the
famous Apollonius of Perga, in Pamphylia, who flourished The term Geometry, which comes from two Greek words, yn, from B.C. 247 to 222, at Alexandria, in the reign of Ptolemy the earth, and Herperv, to measure (pronounced ghee, and Euergetes, another king of the same Ptolemaic dynasty, and met-rine), literally signifies land-measuring, and was originally who was called by his contemporaries the “Great Geometer." applied to the practical purpose which its name signifies, in the He wrote several books, full of discoveries, on the higher land of Egypt, the cradle of the arts and sciences. Herodotus, geometry, and greatly extended the domains of the plane the oldest historian, with the exception of Moses, whose works geometry. Other geometricians of eminence arose in the school have reached us, gives the following account of its origin :-“I of Alexandria, and bequeathed the precious remains of their as informed by the priests at Thebes, that King Sesostris genius to happier times. Claudius Ptolemæus, the author of made a distribution of the territory of Egypt among all his the great work on astronomy called Megale Syntaxis, the Great sabjects, assigning to each an equal portion of land, in the Construction, or Almagest; Pappus, the author of the Matheform of a quadrangle, and that from these allotments he used matical Collections; and others, including Theon and his to derive his revenue, by exacting every year a certain tax. In daughter Hypatia, bring us down to the period when the second cases, however, where a part of the land was washed away by Alexandrian library* was burnt by command of the Mohamthe annual inundations of the Nile, the proprietor was permitted medan barbarian, the Saracen Čaliph Omar, in 64v, and to present himself before the king, and signify what had hap- the labour and learning of ages were irrevocably destroyed. pened. The king then used to send proper officers to examine The dark ages supervened, and little was done in the advance and ascertain, by admeasurement, how much of the land had ment of science until the glorious invention of printing, and the been washed away, in order that the amount of the tax to be general revival of literature about the middle of the fifteenth paid for the future might be proportional to the land which century. remained. From this circumstance I am of opinion that The ancient Greek geometry was speedily made known to the geometry derived its origin; and from hence it was transmitted moderns through the medium of translations of, and commen. into Greece.” The existence of the pyramids, the ruins of the taries upon, the writings of the great masters. The Elements temples, and the other architectural remains of ancient Egypt, of Euclid, indeed, were reckoned so perfect, that no attempt supply evidence that its inhabitants possessed some knowledge was made to supersede them; and the only object of writers on of geometry, even in the higher sense in which we now use the geometry in general was to explain his works, and to make term; although it is possible that the geometrical properties of what additions they could to the science, in the same masterly figures necessary for the construction of such works might have style of composition. A host of names of eminent authors been known only in the form of practical rules, without any might be mentioned, who succeeded in establishing the Greek scientific arrangement of geometrical truths, such as are pre- geometry, and in extending its domains. The principal of these, sented to us in the Elements of Euclid.
however, was Dr. Robert Simson, Professor of Mathematics in The word "geometry,” used in its highest and most extensive the University of Glasgow, who flourished in the middle of the meaning, signifies the science of space; or that science which last century. His grand endeavour was to present to modern investigates and treats of the properties of, and relations Europe the Elements of Euclid as they originally appeared in existing among, definite portions of space, under
the abstract ancient Greece. In this he succeeded to admiration, and his division of lines, angles, surfaces, and volumes, without any edition of this great work maintains its reputation to the regard to the physical properties of the bodies to which they present moment. belong. In this sense, it appears to be very doubtful whether In giving our first lessons on geometry, we think it advisable the Egyptians or Chaldeans knew anything of the science. It to follow what seems to have been the natural course of events is to the Greeks, therefore, that we must look for the real in the history of this science. The present advanced state of origin of geometry, as an abstract science. Thales, the Greek our geometrical knowledge was preceded in early times by a philosopher, born 640 B.C., is reported, by ancient historians, species of practical geometry gathered from experience, and to have astonished even the Egyptians by his knowledge of suited to the wants of those who required its application, before this science. The founder of scientific geometry in Greece, any attempt was made to enter very deeply into the study of however, appears to have been Pythagoras, who was born about the theory. The latter was left to the schools of the philoso568 B.C. He discovered the celebrated 47th proposition of the phers and the academy of Plato. Accordingly, we shall precede first book of Euclid's Elements, and various other valuable and our disquisitions on the Elements of Euclid and other geometers, important theorems. He was great also in astronomy, having both ancient and modern, by a short system of practical rules anticipated the Copernican system of the universe. Plato, and easy explanations in this important science ; and we shall another great geometrician, and founder of the academy at endeavour to make the subject both simple and clear by plain Athens, who was born 429 B.C., was the first who made some definitions, suitable diagrams, and palpable demonstrations, advances into what is called the higher geometry. The next after the manner of the French writers on this subject, who same super-eminent in the science of geometry is that of have even in their more elaborate treatises to a great extent Euclid, whose " Elements" has been the principai text-book for abandoned the system of Euclid. leamers during a period of more than 2,000 years. He fourished at Alexandria, in Egypt, about 300 B.C., during the reign of Ptolemy Lagus, who was one of his pupils, and to has three dimensions, viz., length, breadth, and thickness.
1. Extension, or the space which any body in nature occupies, whom he made the celebrated reply, when asked if there was a This is Euclid's definition of a geometrical solid. shorter way to geometry than by studying his Elements:--"No, sire, there is no royal road to geometry.”
2. A point is the beginning of extension, but no part of it; The prince of ancient mathematicians, however, was the hence it is said to have position in space, but no magnitude. celebrated Archimedes, born at Syracuse B.C. 287, about the to have length without breadth. Hence, also, the extremities
3. A line is extension in one direction only; hence, it is said period of the death of Euclid. His discoveries in geometry, of a line are points ; and lines intersect or cross each other mechanics, and hydrostatics form a remarkable era in the history of the mathematical sciences : and even the remains of
only in points. his works which are still extant constitute the most valuable
4. A straight line is said, by Euclid, to be that which lies part of the ancient Greek geometry. He was the first who evenly between its extreme points ; and, by Archimedes, to be attempted to solve the celebrated problem of the rectification of definitions are defective; the defect is supplied thus: A straight
the shortest distance between any two points. Both of these the circle—that is, finding a straight line exactly equal to the cir- line is such, that if any two points be taken in it, the part cumference. He found out the beautiful ratios of the cylinder which they intercept (or which lies between them) is the to its inscribed sphere and cone, and the quadrature of one of shortest line that can be drawn between those points. the conic sections. His discoveries in physics, or natural are simple, true, and beautiful. The story of the
5. A crooked line is one composed of straight lines joined at determination of the specific gravity of the golden crown of his
* The first library, which was founded by Ptolemy Soter, and which consin, Hiero, King of Syracuse, is well known; and the very was said to have contained 400,000 manuscripts, was accidentally natural shout of * Eūpnka, cüpnka” (pronounced heu-reé-ke), 1 burnt 47 B.c., when Alexandria was taken by Julius Cæsar. The kave found it, I have found it! on coming out of the bath, has second library is supposed to have contained 700,000