LESSONS IN LATIN.—XVII.

THE NUMERALS.

As in English, so in Latin, the numerals have various forms. Thus we say, one, first, one each, once. One, and the corresponding two, three, etc., we call cardinal (from cardo, -inis, m., a hinge), because they are the chief numbers, those on which the others hinge. First, second, third, etc., we call ordinals (from ordo, -Ĭnis, m., an order or series), because they show the order or place in a series which a person or thing holds. One each, two each, three each, etc., may be called distributives, because they distribute the numbers severally to persons or things, declaring how many each possesses. Once, twice, thrice, etc., are called adverbial numerals, because they imply some verb, and state how often the action of the verb takes place. You may see a full view of the Latin numerals in all their forms in the ensuing table, with the English figures on the left hand, and the corresponding Latin characters on the right hand.

Distributive,

one.

first.

one each. singuli, one each.

once.

2. duo, two.

secundus, second, bini, two each.

semel, once. I. bis, twice. II. III.

quater, four times.

IV.

N.

unus

una

unum.

G.

unius

unius

unius.

Where specially observe that the genitive ends in ius, and the dative in i. I put together in a form easy of remembrance the words declined like the preceding example:

Pronouns that make ius in the genitive and i in the dative.

unus, ullus, nullus.

solus, totus, alius. uter, alter, neuter.

Observe that the ius and the i of the masculine gender remain

So in their compounds: by adding que, uter becomes uterque, which signifies both, and is formed thus: nom. uterque, m., utrăque, f., utrumque, n.; gen., utriusque; dat., utrique; so, nom., unusquisque, m., unaquæque, f., unumquodque, n.; gen., uniuscujusque; dat., unicuique. Alius in the neuter, has aliud; in the genitive singular, alius (contracted from aliius), and dative alii. In alteruter, one of two, commonly uter alone is declined thus:

N. G.

alteruter, m. alterutrius

alterutra, f. alterutri

alterutrum, n. alterutrius.

Declension of duo, duæ, duo, two.

M.

N. & V.

duo

F.

duæ

duārum

duabus

duas

N. duo.

duōrum.

duobus.

duo.

Tres, m., tres, f., tria, n., three.

21. unus et vi- unus et vicesi- viceni singuli, vicies semel. XXI. ginti.

mus.

Acc. Milia is declined like tria, thus: milia, milium, milibus, milia. Milia requires after it a genitive; for instead of saying, as we do, ten thousand men, the Latins said, decem milia hostium, ten thousands of men; but mille considered as a whole, a thousand, is indeclinable: thus, dux cum mille militibus, a general with a thousand soldiers. The ordinals are declined like nouns of the first and the second declension. The distributives are also declined after the same manner. Mark that singuli is in the plural. The plural is necessitated by the meaning, inasmuch as the adjective is a distributive, for distribution implies more than one; thus the Latins said, inter singulos homines, among the men severally.

If now you carry your eye down the numbers, you will find that for every separate number from one to nine, there is a separate word. With ten (decem) a new series begins, which goes on to nineteen, when again at twenty (viginti) a new word begins a new series. In centum and in mille, you also find new

to the utmost extent that the limits of that figure will permit; and in the third the prefix circum gives the word the meaning of drawing one figure round another. Inscription and circumscription indicate operations that are precisely the reverse of each other.

PROBLEM XLII.—In a given circle to inscribe a triangle equiangular to a given triangle.

X

A

equiangular to

Let A B C (Fig. 61) be the given triangle, and D E F the given circle: it is required to inscribe in the given circle DEF a triangle the given triangle A B C. At any point D in the circumference of the circle DEF draw X Y as a tangent to the circle, and at the point D in the straight line x Y make the angle Y D F equal to the angle ABC, and the angle X D E equal to the angle A C B; and let the straight lines DE, D F cut the circumference of the circle D E F in the points E and F. Join E F. The triangle D E F, inscribed in the circle D E F, is equiangular to the given triangle A B C.

E

Fig. 61.

B

If it is desired to cut off a segment of a circle that shall contain an angle equal to a given angle, as in the above figure to cut off from the circle D E F a segment that shall contain an angle equal to the angle A B C, all that we have to do is to draw a tangent to the circle, and at the point of contact make an angle equal to the given angle, as the angle Y DF was made equal to the angle A B C. The leg DF of the angle Y D F must then be produced far enough to cut the circumference of the circle D E F in the point F. Any angle that may then be formed by drawing straight lines from D and F to any point in the segment, as the angle DEF or the angle D G F, is equal to the given angle A B C.

the point E, the point of intersection of the straight lines a C, B D, at the distance E A, E B, E C, or E D, describe the circle A B C D. This circle touches the sides of the given square F G H K, and is inscribed within it, as required.

To circumscribe a circle about the given square FGH K, find the point E as before, and then from the point E as centre, with a radius equal to the straight line joining E with any one of the four corners of the square, describe the circle F G H K. The circumference of the circle F G H K passes through the other three corners of the square, and the circle F G H K is therefore circumscribed about the square F G H K, as required.

F

H

Fig. 63.

We may now pass on to the construction of regular polygons. The term polygon is derived from two Greek words, woλus (pol-'use), much or many, and ywvia (go'nia), an angle, and means a figure that has many angles. Many-angled figures are also called multilateral or many-sided figures, from the Latin multus, much or many, and latus, a side. 'Polygon" and "multilateral figure are terms which may be considered to mean precisely the same thing, for a figure that has many angles must, of course, have many sides. It has, in fact, just so many sides as it has angles, and the most familiar illustration of this that can be given is that of a room, which, generally speaking, has just four sides and four corners or angles. The terms " polygon" and "multilateral figure are applied, as we have been taught in Definition 33 (Vol. I., p. 53), to any figure that has more than four sides. A polygon may be regular or irregular-that is to say, its sides and angles may be equal or unequal. The student has already been shown the method of making triangles equal to given irregular polygons; and the construction of an irregular polygon of any number of sides, having its angles equal to

Let A B C (Fig. 62) be the given triangle, and DEF the given circle about which it is required to circumscribe a triangle equiH angular to the given triangle

D

Y

А В С.

Produce B C, the base of the triangle A B C, both ways to x and y. Draw K L touching the circle D E F in the point E, and from the centre G of the circle D E F draw the straight line GE perpendicular to K L. Then at the point G, in the straight line E G, make the angle EGF equal to the angle ACY, and the angle E G D equal to the angle A B X. Through the points D and F draw the straight lines H K, H L, meeting each other in the point H, and the straight line K L in the points K and L. The triangle HKL circumscribed about the circle D E F is equiangular to the given triangle A B C.

Fig. 62.

PROBLEM XLIV.-To inscribe a square in a given circle, and about the same circle to circumscribe a square.

Let A B C D (Fig. 63) be the given circle, and E its centre. Through E draw the diameters A C, B D at right angles to each other, and join A B, B C, C D, and D A. The figure A B C D thus formed is a square, and it is inscribed in the given circle A B C D, as required.

To circumscribe a square about the circle A B C D, draw the diameters A C, BD as before. Through the points A and c draw the straight lines F G, H K parallel to B D, and through the points B and D draw the straight lines F K, G H parallel to A C. The figure F G H K thus formed is a square, and it is circumscribed about the circle A B C D, as required. PROBLEM XLV.-To inscribe a circle in a given square, and about the same square to circumscribe a circle.

Let F G H K (Fig. 63) be the given square: it is required to inscribe a circle within the given square F G H K, and to circumscribe a circle about it. First bisect the sides F G, F K in the points A and B, and through A draw a C parallel to F K or GH, and through в draw B D parallel to F G or HK. From

could readily accomplish, provided that he has paid sufficient attention to our lessons to understand thoroughly all that we have advanced. It is with the construction of regular polygons only that we have now to do.

In Definition 34 we were further taught that polygons are divided into classes according to the number of their sides and angles. Some of these classes have no distinctive name, as will be seen from the following table; but many of them have a name by which the number of their sides can be recognised at once. Thus the polygon that has five sides and five angles is called a pentagon, from the Greek Teνтe (pen ́te), five, and "ywvia, an angle; the polygon that has six sides and six angles is called a hexagon, from the Greek (hex), six, and ywvia, an angle; and so on, the Greek or Latin word for the number of the sides, or some modification of it, being prefixed to the termination gon. A triangle would be called a trigon, and a square a tetragon, according to this system of naming figures from the number of their angles.

The number of degrees in the angle of any regular polygon may be found arithmetically by the following process:

The angles formed by any number of lines meeting together in a point, such as the lines drawn from the angles of any polygon to any point within it, or, in the case of a regular polygon, to its centre, are together equal to four right angles, or 360 degrees. The greater the number of sides of any regular polygon, the less will be the angle at its centre, subtended by each of its sides; and to find the number of degrees contained in its opening, we have B to do nothing more than to divide 360 by the number of sides. For example, in the regular pentagon, or five-sided figure A B C DE, in Fig. 64, it is clear that each of the five angles, A F B, B F C, CF D, D FE, E FA, formed by drawing straight lines from its five salient angles at A, B, C, D E, to its centre, F, is equal to one-fifth of 360 degrees, or, in other words, is an angle of 72 degrees. We now wish to find the numerical value of any and all of the angles of the polygon in degrees. We know that the three interior angles of any triangle are together equal to two right angles, or 180

Fig. 64.

D

degrees; therefore, the three interior angles of any of the five equal and equiangular triangles, of which the pentagon in Fig. 64 is made up, are together equal to 180 degrees. Now the angle at the apex, F, of any of these angles was shown to be equal to 72 degrees, therefore the angles at the base are together equal to 180-72 degrees, or 108 degrees. But as the triangles which compose the pentagon are isosceles triangles, the angles at the base are equal to each other, and each of them contains 1082, or 54 degrees. Any angle of the polygon, which is, of course, composed of two of these equal angles, contains 108 degrees. The following is a table of regular polygons, from the triangle or trigon of three sides and angles to the polygon of twenty sides and angles, with the numerical value of the angle of each polygon in degrees, minutes, seconds, and fractional parts of a second; and the numerical values of the angles at the apex and base of the triangles into which each polygon may be divided by drawing straight lines from its salient angles to its centre.

TABLE OF REGULAR POLYGONS.

LESSONS IN MUSIC.-IX.

CHARACTER AND EFFECT OF PRINCIPAL NOTES. DOH (being the "governing" note) gives a sense of POWER to the hearer, and of SECURITY to the singer in a greater degree than any other. The singer feels it to be the note to which he can, from any point, most easily return. It is more easy to perceive musical effects, than to find words that will sufficiently represent them; but if the names must be given, this note should be called (in reference to its effect in a slow movement) THE STRONG or firm NOTE.

Soн has a similar effect to DOн, but is not equal to it in power. It may be distinguished (when sung slowly) as THE GRAND or clear NOTE.

ME has a somewhat graver and softer effect than Soн. It may be denominated (especially in slow movements), THE STEADY or calm NOTE.

When DOH, ME, and soн predominate in a tune, they contribute to its general character: if the movement be a quick one, great BOLDNESS and DECISION; and if the movement be a slow one, they give it DIGNIFIED SOLEMNITY. Of course, the power of any particular note to give a character to the tune in which it occurs, will depend on the frequency and the emphasis with which it is used, and will be modified, also, by the kind of "measure and the rythmical arrangement in which it occurs. In studying the following examples, let the pupil always strike or sing the chord of the key-note, and part, at least, of the scale before he begins to sing the phrase; for our assertions in reference to the mental effects of notes are not true, unless the ear is first filled with "a sense 99 of the scale in which they

occur.

For "dignified solemnity," notice the power of гOH, ME, and SOH in the following opening phrases from Handel. You will remark how, in each case, the great artist takes advantage of these bold and grand notes to bring out, by contrast, a very

But another striking illustration occurs to us. A minister had heard, and had been greatly moved by Mendelssohn's song"Oh! rest in the Lord." He preached on the text, and thought much of that repose which comes, not with weariness or sleep, but with living blessedness. Some time after, thinking on the power of the notes of the scale, it occurred to him that Mendelssohn must surely have expressed the idea of rest in God, chiefly by means of the third note of the scale, ME, which we have called the note of serene repose. "If by any other note," he said to himself, "that peculiar effect was produced on my mind so strongly, then the theory about mental effects must fall to the ground." He at once analysed the song, and found that the very first emphatic note was that which he had expected that the power of this note was brought out, by placing it in ever-varying but most effective positions; and that, even when the key changes, the ear is surprised and pleased by the recurrence again of this same third note in the new scale. Among other studies in this delightful song, it was pleasant to notice the change in the manner in which the word "Lord" is expressed in the latter part of the song. At first, it is uttered with the firm and sure confidence of the note DOH; but, when that spirit of confidence has risen to a somewhat triumphant feeling, then it must use the "clear" and "grand' note soн.

Notice the effect of ME, in each case where it occurs on the strong parts of the measure, but especially in the last case in this quotation. What full-hearted satisfaction and perfect rest it brings? The words and the music aid each other to move the heart.

For the effect of "boldness and decision," which DOH, ME, and SOH (sung somewhat quickly) give to a tune, we may quote the martial music of Handel. "See the conquering Hero comes" begins thus:

KEY G.

THE word "tone," in elocution, may be used, as in music, to signify the interval which exists in successive sounds of the voice, as they occur in the gamut, or musical scale. But it is commonly used as equivalent, nearly, to the term "expression" in music, by which is meant the mode of voice as adapted, or not adapted, to feeling. Thus we speak of the "tones" of passion-of a "false" tone-of a "school" tone.

66

66

very

Every tone of the voice implies-1, a certain "force," or quantity," of sound; 2, a particular "note," or "pitch;" 3, a given "time," or "movement;" 4, a peculiar "stress;" 5, a special "quality," or character; 6, a predominating "inflection." Thus, the tone of awe has " a very soft force," a low pitch," a very slow movement," "medial stress," and "pectoral quality," or that deep murmuring resonance which makes the voice seem, as it were, partially muffled in the chest, together with a partial "monotone," prevailing at the opening of every clause and every sentence. All these properties belong to the natural utterance of awe; take away any one, and the effect of emotion is lost-the expression sounds deficient to the ear. [xx]* Example 1. The bell strikes❘ one. We take [oo] nō nōte of time,

[=] But from its loss: to give it, then, a tōngue, [m.s.] Is wise in man. As if an àngel | spoke || P.q. I feel the solemn sound. If heard aright,

It is the knell of my departed hours. Where are they ?-With the years beyond the flood. The first five of the properties of voice which have been enumerated, are the ground of the following classification and

notation::

KEY TO THE NOTATION OF EXPRESSIVE TONE."
"Force."

[1] "loud;" [I]" very loud;" [x] "soft;" [xx] "very soft;" [<] "increase;" [>] "decrease."

These marks indicate [xx] "very soft;" [oo] "very low" [=] very slow;" [m.s.]" medial stress; " [p.q.] "pectoral quality."-Se Key to the Notation of “Expressive Tone.”