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LESSONS IN GREEK.-No. XXIX

By JOHN R. Beard, D.D.

THE AUGMENTS.

The Augment and the Reduplication.

We must now give fuller particulars respecting the Augment. The Augment is specifically the token of past time. Consequently, it forms a part of the historical tenses-namely, the Imperfect, the Pluperfect, and the Aorist; but it is retained in no other mood than the indicative. The Perfect, though a principal tense, takes a Reduplication, and to this reduplicated form an Augment is prefixed to form the Pluperfect. The Augment, considered as distinct from the Reduplication, appears in two forms of these, one is called the Syllabic; the other, the Temporal.

The Syllabic Augment.

The Syllabic Augment is an e, which in verbs whose root begins with a consonant is prefixed to the stem of the Imperfect and the Aorist, and to the reduplication in the Pluperfect. Thereby is the word augmented (hence the name) by one syllable in the Imperfect and the Aorist, and by two syllables (including the reduplication) in the Pluperfect, e. g. Xvw, Imperfect ε-λυον, Aorist ε-λυσα, Pluperfect ελελύκειν.

When the root begins with p, the p is doubled before receiving the Augment, as pirrw (I throw), Imperfect EppiπTOV, Aorist ερριψα, Pluperfect ερριφειν.

In the three verbs, βούλομαι, I will, δυναμαι, I am able, and μew, I intend, the Augment with the Attics, the later rather than the earlier writers, is n instead of ; as Impf. ηβουλομην as well as εβουλόμην, Aorist ηβουλήθην as well as εβουλήθην, Impf. ηδυνάμην as well as εδυνάμην, Aorist ηδυνη θην as well as εδυνηθην (instead of εδυνάσθην), Impf. ημελλον as well as εμελλον : the Aorist is very seldom ημελλησα.

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In alpew, ppeor, the a is lengthened into n, and the is subscript; thus, p. In avλew, nuλtov, the a is simply lengthened into the n. In oIKTILw, WKrilov, the o is lengthened into w, and the is subscript; thus, q.

The Augment is unattended with any change in the verbs which begin with ŋ, i, v, w, ov, and , inasmuch as the initial syllables are already long; as Traoμaι, I submit (I am worse, inferior), pf. ήττημαι, plpf. ἡττημην, ἵποω, I press upon, aor. έπωσα; ύπνοω, I put to sleep, aor. ύπνωσα; ωφελέω, 1 benefit, impf. ωφελεον; ουτάζω, I wound, impf. ουταζον ; είκω, I yield, impf. ekov, aor. tika: ekazw, I liken, guess, forms an excep tion, which, though but seldom, changes the inton and underwrites the 4, thus giving rise to these two forms, eukalov, γκαζον; εικασα, γκασα ; εικασμαι, γκασμαι.

Those verbs also are commonly without the augment whose

root begins with ev; e.g. evxoμai, I pray, evxoμny, less often ηυχομην ; but the Pluperfect is ηυγμαι, the e being augmented inton: Eupiokw, I find, discover, in good prose rejects the aug

ment.

Verbs which begin with ă and a following vowel have in the augmented form a instead of n, as aïw (a poetic word), I feel or apprehend, impf. aïov: in those which begin with ā, av, ot, and a following vowel, there is no change for the augment, as impf. avaivov; outsiw, I steer, impf. oiakilov: also avadiokw, αηδιζομαι, I am displeased, impf. αηδιζομην ; αυαίνω, I dry up, I destroy, though no vowel follows the a, has avaλwoa, avāAwka, and also avλwoa and avλwka. However, the poetic atdw (in prose gow), I sing, and aïoow (Attic goow), I rush, take the augment, as netdov (in prose yoov), niža (Attic pa). Otopar, I think, impf. wouny, belongs not here, because the o following the or is not a part of the root.

οι

with ot and a following consonant, as oikoupew, I govern a house, There is no augmental change also in some verbs beginning aorist οικουρησα; οινίζω, I desire wine, impf. οινιζον; οινοω, I indulge in wine, pf. mid. or pass. ovoμevoc and also qvwμενος ; οιστράω, I madden, aor. οιστρησα.

ment es instead of ŋ: The following verbs beginning with have for their aug

saw, I permit, impf. εiwy, aor. elaσa.

iw, I accustom, aor. tioa, pf. εwba, I am accustomed: eiσa (from the stem ɛd), I establish, is poetic; in prose there are only abauevos and coauevos, grounding, instituting. Moow, I wind, roll, perf. mid. or pass. eiλtyμai. Xxw, I draw, drag, aor. eikvoa (stem, iλkv), tiλov (stem iλ), I took, commonly called the Aorist of aipew, I choose, take. έπομαι, I follow, impf. είπομην. εργαζομαι, I labour, pf. ειργασμαι. prw (prνw), I creep, aor. 2. EiOTOV. ioriaw, I entertain a guest, pl. eioriāka εχω, I have, aor. 2. είχον.

The ensuing verbs take the syllabic augment instead of the temporal; namely,

ayvμ, I break, a. saka; perf. 2. ɛaya, I am broken. άλισκομαι, I am being caught, pf. εαλωκα, also ήλωκα, I am caught (captus sum).

avdavw, I please, impf. lavdavov, pf. iāda, aor. 2. ¿ãdov. oupew, I make water, impf. toupouv, pf. εovρnra. whew, I push, impf. ewlovv; sometimes without augment, as διωθουντο.

ωνεομαι, I purchase, impf. εωνουμην, also ωνούμην.

The verb opralw, I celebrate a festival, takes the augment in the second syllable, as impf. ἑωρταζον. This happens also in I resemble, pf, 2. ɛoɩka, I am like; ɛoɩxɛ, it is likely; plpf.

είκω,

εκειν.

πειν } poetic.

έργω, I do, pf. 2. εοργα, plpf. ἑωργειν ελπομαι, I hope, pf. 2. εολπα, plpf. εωλπειν

The following three verbs have both the syllabic and the temporal augment; the aspirate of the root passes to the augment &:

opaw, I behold, impf. ¿wowv, pf. iwpara, iwpapai. ανοιγω, I open, impf. ανεωγον, a. ανέωξα (intn. ανοιξαι). άλισκομαι, I am being caught, a. ἑλλων, (infn. ἁλωναί,) also ήλων.

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The Attic Reduplication.

Several verbs beginning with a ore or o repeat in the Perfect and the Pluperfect, before the temporal vowel, the two first letters of the stem : this augmentation is called the Attic Reduplication. The Pluperfect very seldom takes a new augment, as διωρωρυκτο, from ορυσσω ( I dig), fut. ορύξω, pert. ορ-ώρυχα, perf. pass. or mid. ορωρυγμαι, plpt. opand ωρ-ωρυγμην : in ηκ-ηκοειν the Pluperfect is regular. The Temporal Augment, as well as the Reduplication, remains in all the moods as well as in the participle. The Attic reduplication affects verbs of two classes: 1. Verbs whose stem-syllable is short by nature, e. g. προω, I plough,

ελεγχω, I convince,

ελαω (ελαύνω), I drive, ορυττω, I dig,

αρτηρόκα, αρ-ηρομαι αρ-ηρόκειν, αρ-ηρομην ελ-ήλεγχα, εληλεγμαι ελ-ηλέγχειν, ελ ηλεγμην ελ-ήλακα, εληλαμαι ελ-ηλακειν, ελ-ηλαμην ορωρυχα, ορωρυγμαι ορωρυχειν, ορωρυγμην syllable of the stem have a

I cast together,

I arise in, εμ-βαλλω,

I throw in, συσκευάζω, I pack up,

συν-ερριπτον συνέρριφα συν-ερρίφειν

ενεβαλλον εμ-βεβληκα εν εβεβλήκειν

συν-εσκευάζον συν-εσκευακα συν-εσκευακεί

The Student should go carefully over this table and account for every change which it presents. I give an example or two. In αποβαλλω, the o of the preposition is dropped before the vowel of εβαλλον, to prevent the hatus or gaping occasioned by two vowels coming immediately together; but as in βεβληκα the reason ceases, so the o is resumed, and you have αποβεβληκα; yet again απ-εβεβλήκειν. In συλλέγω the the verb has changed the of the preposition into its own sound, namely, λ; but when the preposition is not immediately subjected to the form of the A, it resumes its own, as in συνελεγον.

of

Verbs which are made up of ους, hardly, with difficulty, take the augment of the reduplication, 1. in front, or at the beginning, when the root of the simple verb begins with a consonant or with η or ω ; and 2. in the middle, when the root of the simple verb begins with any other vowel except ŋ and w;

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2. Verbs which in the second vowel long by nature, which after prefixing the augment they δυς-αρεστεω, shorten; except ερειδω, I support, stem ερ-ηρεικα, ερ-ηρεισμαι.

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I displease,

ε-δυς-ωπουν δε-δυς-ωπηκα εν δε δυς-ωπήκειν δυς-ηρεστούν δυς-ηρίστηκα δυς-ηρεστήκει

These two laws are observed by compounds of ev, well; only that such compounds avoid the augment at the begin

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EXERCISES, GREEK-ENGLISH. N.B. Tell the part and give the English of each of these forms :

Ηνωρθουν; επαρώνουν; ήνωχλησα; ηνωρθωκα; εδιηκόνεον; διηταόμην ; ηνειχομην ; εμυθολογουν; ᾠκοδομηκα ; ερριπτον ; ηγον; ηλπικά; ἱκετευκα; ώμιληκα; φκτικά ; ρεον; ευχόμην, αναλωσα; ειων; είλκυσα; ειπομην; εἰστιᾶκα; είχον; ἑαλωκα ;

Verbs derived from compound nouns or adjectives take the ἑάλων; τεθυκα; εγεγραφεῖν; τεθλακα, εγλυφα, εγνωρίκειν ; augment at the beginning; e. g.

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The analogy of these verbs is followed by two other verbs which are not formed with the aid of prepositions, but by derivation from other compounds; e. g.

διαιταω (from διαιτα. subsistence), I feed, impf. εδιήταον and διήταον, aor. ἐδιῄτησα and διήτησα, pf. δεδιψτηκα ; mid. διαιταομαι, I live, διηταύμην. διακονέω (from διάκονος, a servant, our deacon), I serve, impf. εδιηκόνεον and διηκόνεον, pf. δεδιηκονηκα.

As exceptions, some verbs compounded with prepositions take the augment before the preposition: these are verbs in which the preposition and the verb have so coalesced as to present the signification of a simple verb; e. g. αμφιγνοεω (νοεw, I think), impf. ημφιγνοεον

I am in doubt,

αμφιεννυμι,

I put on, clothes

επισταμαι,

I understand,

αφιημι,

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aor. ημφιεσα, pf. m. or p. ημφιεσμαι impf. ηπισταμην

εξένωκα ; εκτίκειν ; συνειλοχα ; ειλήφειν ; λελεγμαι; ορωρυγμαι ; εληλεγμαι; αληλιμμαι ; ηκηκόειν ; εγήγερμην ; συνελεγον ; συνερριφα; απεβαλλον; προυβαλλον ; εγγεγονα; συνεσκευάζον; δυςηρεστούν ; ευεργέτηκα; μεμυθολογηκα.

You must not only give the English and assign the part (mood, tense, &c.), but explain the formation of each word, are produced, and the rule or remark which the formation giving the derivation, the manner in which the several parts exemplifies, as set forth in what precedes. The task is not an easy one, and you will be tempted to pass it over as unnecessary. But if you satisfy yourself with a general view of the matter in your first study of this manual, fail not to return to this part and all the harder parts, and go over them again and again, until you have mastered them. Depend on it, you only multiply your difficulties by passing slightly over the harder and less attractive instructions. Nothing is here given but what is necessary to a correct and complete acquaintance withTM Greek prose; and if you wish to know the language, you must, sooner or later, acquire these details; and from long experience in learning and teaching, I can assure you that the sooner you master them, the better.

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Logarithms: 5.000868 4.000868 3.000868 2.000868 1.000868 0.000868

εκαθίζον, pf. κεκαθικά

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εκαθέζομην and καθεζόμην

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44. The preceding tables and remarks clearly show the advantages over every other, which the common system of logarithms possesses, in consequence of its being the same as the root of the decimal scale of notation. By merely increasing or diminishing by unity the index of a logarithm of a number, the logarithm of a decimal multiple or submultiple of that number is immediately obtained. Hence, the calculation of the loga. rithm of one number is sufficient for the determination of innumerable others ; for, by tabulating the decimal parts of the logarithms of all integers from 1 to 10,000, or from 1 to 100,000, etc., the complete logarithms of such numbers can easily be found, whether they be considered as integers, decimals, or mixed numbers; the proper mdices being supplied according to the foregoing rules.

45. A system of legarithms founded on any other base but 10,

1

mod. com. system;

Nep. log. 10

would want all the advantages above-mentioned. The loga-system: modulus of common system; but common logarithm of rithms of all such numbers as are determined by the mere 10-1; therefore, by the rule of Proportion, the fourth term is change of the index in the common system, would require to be separately calculated and tabulated with their indices. The logarithms of all fractions, as well as integers, and the logarithms of all numbers of which the factors were powers of the base, would require the same operation to be performed. that is, the modulus of the common system of logarithms is the For though, in the latter case, the calculation of the logarithms reciprocal of the Neperian logarithm of its base. would be as easy as before, yet their tabulation with indices 52. The Logarithmic Series is analytically exhibited in a would still be necessary, as the bare inspection of the numbers variety of curious forms. The following rule, which is a themselves would not be sufficient to suggest the properverbal translation of one of the most useful of these forms, change to be made on the indices, as in the common system. The disadvantages of such a system would even be more strongly felt in the reverse operation of finding from the tables the numbers corresponding to any given logarithms.

46. In addition to the decimal parts of the logarithms of the common system, which are given in Tables of Logarithms, the average differences of every five logarithms are usually given in an adjoining column, for the purpose of rendering it easy to obtain the approximate logarithms of numbers greater than those contained in this table. The approximate logarithms of such numbers are obtained on the principle, that the differences of numbers which differ little from each other, are nearly proportional to the differences of their logarithms. Thus, in Part I. of the Skeleton Table, Art. 31, the successive differences of the numbers 1'00056, 1.00028, and 1.00014, are 00028 and 00014; and the differences of their logarithms are 000122 and 000061; now, the following proportion is correct, as far as the decimals extend :

00028 00014:: 000122 : 000061.

But were the decimals further extended, this proportion would be found to be only nearly correct. The application of the principle thus established, however, is sufficiently correct for all practical purposes.

NEPERIAN SYSTEM OF LOGARITHMS.

47. The system of logarithms, first invented by Napier, and sometimes, but improperly, denominated the Hyperbolic, is, theoretically speaking, the most natural. The base of this system, which is easily deduced from an analytical formula called the Exponential Theorem, is 2.718281828459, etc.; this number, however, can only be accurately expressed by the following infinite series:

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24 120 720

+ etc.

48. The mathematical construction of logarithms depends on an analytical formula, denominated the Logarithmic Series, in which it is shown that the logarithm of a number in every system can be expressed by the same infinite series, united to a factor called the Modulus, which is a constant function, or invariable modification of the base. This series will be given under the head of Algebra.

49. In the Logarithmic Series, the modulus is such a function of the base, that if an integer be assumed as the base of a system, the modulus of that system becomes an infinite series, as in the common system; and if an integer be assumed as the modulus, the base becomes an infinite series, as in the Neperian system.

may be employed in the construction of a table, either of Neperian or of common logarithms. It is universally applicable, and possesses this valuable property, that the infinite series converges with greater rapidity, in proportion as the given number increases in magnitude.

53. To find the Neperian, and thence the common, logarithm of a given number, the Neperian logarithm of the difference between that number and unity being given. Rule: Divide unity by the difference between double the given number and unity, for a first quotient; divide this quotient by the square of that difference, for a second quotient; divide the second quotient by the same square, for a third quotient; divide the third quotient by the same square, for a fourth quotient; and so on. Divide these quotients respectively by the odd numbers in the series 1, 3, 5, 7, 9, 11, 13, etc.; that is, divide the first quotient by 1; the second by 3; the third by 5; the fourth by 7; and so on. Find the sum of as many of the latter quotients as have significant figures two or three decimal places beyond the extent to which the logarithms are required to be accurate; then, to double this sum, add the Neperian logarithm of the difference between the given number and unity, and the result is the Neperian logarithm of the given number. Lastly, multiply this logarithm by the modulus of the common system of logarithms, and the product will be the common logarithm of the given number.

54. Example 1. To find the Neperian logarithm of the number 2. Subtract unity from 4, which is double the given number, and divide unity by the remainder 3; then, divide this quotient by the square of 3, which is 9, and so on, as in the following operation:

Dividends.

-

Quotients.

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Sum 34657359

To double this sum, which is 69314718, add the Neperian logarithm of 1, the difference between the given number and unity; in this case the result will still be 69314718; for the logarithm of 1 is 0, in the Neperian, as in every other system. Therefore, 69314718 is the Neperian logarithm of the number 2.

50. The peculiarity which distinguishes the Neperian system of logarithms from every other, consists in the simplicity of its modulus, which is unity. By the adoption of this modulus, the logarithms in this system are evidently rendered independent of the base; hence it is called the most natural. This remark shows that it was possible for Napier, the original inventor, to construct his logarithms without reference to any assumed number as a base. The wonder still is, how he made the discovery half a century before the Logarithmic Series was known. 51. From Art. 48, it is easily seen that the logarithms of the same number in different systems are proportional to the moduli of those systems respectively. Hence, the modulus of 56. Example 3. To find the Neperian logarithm of 5. Divide the Neperian system being unity, the modulus of the common unity by 9, the difference between double 5 and unity; then system is found by the following proportion :-As Neperian divide the quotient by 81, the square of 9, and so on, as follogarithm of 10 common logarithm of 10:: modulus of Neperian | lows:

55. Example 2. To find the Neperian logarithm of the nurnber 4, which is the square of 2. Multiply 69314718 by 2, the index of the square, and the product 1-38629436 is the Neperian logarithm of 4 (Art. 21).

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behind their countrymen in appreciating the value of the preceding Lessons on Arithmetical Logarithms. About a year after the publication of the work first mentioned, we put a copy into the hands of a plain country highlandman from the Hebrides, or Western Isles of Scotland, where he had been born and brought up all his days. Some time after, we

These quotients are now to be divided by the series of odd received a letter, from which the following are extracts; and numbers, as follows:

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To double this sum, which is 22314355, add 1.88629436, the Neperian logarithm of 4, the difference between the given number and unity, and the result 1-60943791 is the Neperian logarithm of 5.

we give these extracts in the hope that it will encourage many of our students to go and do likewise.

"You will remember that you had the kindness to present to me your very valuable manual upon the Construction of Logarithms; a thing I had been in quest of for seventeen or eighteen years past; and having, in vain, consulted many authors upon this subject, I, in despair, had given it up, as a thing far above my narrow comprehension. About August last, however, having, as deeply as I could, fallen upon the study of your Rules, I was very pleasantly disappointed of failing as before; instead of this, your admirably-handled rules diffused such a flood of light upon my mind as will for ever (while I enjoy the use of my reason) dispel those clouds of darkness that for many years hovered over my understanding

57. Example 4. To find the Neperian logarithm of 10, add together the Neperian logarithms of 2 and 5, and the sum 2.30258509 is the Neperian logarithm of 10 (Art. 18). Conse-respecting the noble logarithms. In order that you may have quently, the reciprocal of this number, which is 4342944819, is the modulus of the common system of logarithms. 58. Example 5. To find the common logarithms of 2, 4, and 6. Multiply the Neperian logarithms of these numbers by the modulus of the common system, and the products will be the common logarithms of the numbers, as follows:Numbers. Neperian Logs.

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Modulus. Common Logs. 0.69314718 X 4342944819 = 0.301030 1.38629436 X 4342944819 0.602060 1-60943791 X 4342944819 = 0.698970 59. The process of multiplication indicated above may be reduced to that of simple addition by employing the following Table, which will be found very useful in the construction and conversion of logarithms :

TABLE OF DIGITAL MULTIPLES.

OF THE COMMON MODULUS AND ITS RECIPROCAL;
Their values to 30 decimal places, being as follows:
M0-434294481903251827651128918917
1: M = 2·302585092994045684017991454684

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The preceding arithmetical illustrations of the method of finding the logarithms of numbers, is principally taken from a work written and published by the editor, at Glasgow, in 1834, entitled "The Mathematical Calculator, or Tables of Logarithms of Numbers, and of Logarithmic Sines and Tangents; with other useful Tables and an Introduction Theoretical and Practical." This work, in its original form, has been long out of print. Subsequently, the tables were published, along with their description, in the form of "The Practical Mathematician's Pocket Guide;" and afterwards, the Theoretical and Practical Introduction, in the form of "The Universal Calculator's Pocket Guide." The author has the satisfaction to think that his simplification of this abstruse subject has been well received by the public; as both books, but especially the Tables, are indeed the pocket companion of many thousand workmen, particularly in her Majesty's dockyards, etc. He believes the students of the POPULAR EDUCATOR will not be These works are published in Glasgow, by W. R. Macphun; and in London, by Hall and Co. Paternoster Row, price 1s. 6d. each.

more than a bare statement of what I have said, I herewith send you a Table of Logarithms, which I constructed by your rules, taking the number 2 as the base of the system. This I suppose will not be disagreeable to you, and will serve to convince you that I can construct a Table of Logarithms at pleasure, though with more labour than the learned can do."

In the middle of his letter, the writer here inserts a "Table of the Logarithms" of all numbers from 1 to 100, with their indices, in five columns, calculated correctly and ingeniously to the base 2. We do not consider it necessary to insert this table here, as at the end of this Lesson, or rather in our next number, we shall insert a Table of the Logarithms of all numbers from 1 to 10,000, calculated to the base 10 (that is, the common system), with a description of the manner of using them in arithmetical calculations. We shall also insert a Table of Antilogarithms, by which the numbers corresponding to any given logarithm within the same limits can be found.

Our correspondent then adds, "I also, by your first rule, and by Part I. of your third skeleton table, calculated the common logarithms of all the prime numbers you pointed out in your manual; and by giving eight or nine decimal places to the natural numbers in the table, I found my answers to agree exactly with your Tabular Logarithms; but as for your second rule, and Part II. of your third skeleton table, I could not, by them, find such correct answers as I found by the first rule. Pray, tell me, how I can find a large collection of the logarithms of numbers from unity to hundreds or rather thousands of millions; and what price will be required for the same, as I feel great curiosity to see the logarithms more fully than I have hitherto done; I will now conclude by saying that your very name will, all my lifetime, be dear to me; not altogether on account of your private lessons on the use of the quadrant, but chiefly on account of the discovery which your manual has unfolded to me about the logarithms a thing which you might have kept hid from me, had not your own kindness prompted you to disclose this secret to me.'

We have now disclosed this secret to our students, and we hope they will make as good use of it as our grateful correspondent did, whose letter is dated Jura, 25th February, 1837.

LESSONS IN READING AND ELOCUTION.
No. VII.

ANALYSIS OF THE VOICE.
Ir we observe attentively the voice of a good reader or speaker,
we shall find his style of utterance marked by the following
traits. His voice pleases the ear by its very sound. It is
wholly free from affected suavity; yet, while perfectly natural,
it is round, smooth, and agreeable. It is equally free from
the faults of feebleness and of undue loudness. It is perfectly
distinct, in the execution of every sound, in every word. It is
free from errors of negligent usage and corrupted style in
pronunciation. It avoids a measured, rhythmical chant, on the
one hand, and a broken irregular movement, on the other. It

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