LESSONS IN LOGARITHMS.-III.

COMMON SYSTEM OF LOGARITHMS.

LOGARITHMS OF THE POWERS OF 2.

powers (see Art. 22).
are constructed :-

On this principle, the following tables

Log. 4 = 2 ×
8 = 3 x
164 x

301030 = 602060. Log 32 = 5 x 301030 = 1.505150. 301030 = *903090. 64 6 x 301030 = 1·8 6180. 301030 = 1·204120. etc. etc. etc. LOGARITHMS OF THE POWERS OF 3.

34. To find the Logarithm of any Prime Number.-Rule 1. Divide the given prime number by the natural number nearest to it in the skeleton tables, but less; divide the quotient by the natural number nearest to it, but less; divide this quotient by the natural number nearest to it, but less; and so on, till the last quotient coincide with some natural number in the Log. 9 = 2 x 477121 = tables; then, the last quotient with all the divisors are the tabular factors of which the prime number is composed. Consequently, if the logarithms of all these factors, given in the tables, be added together, their sum will be the logarithm of the given prime number. On this principle the following table, exhibiting the method of calculating the logarithm of the prime number 2, is constructed :

FIRST CALCULATION OF THE LOGARITHM OF 2. Dividends.

Divisors.

2-00000 + 1.77829 = 1.12463 + 1.07461 =

Quotients.
1.12463

1.04660

Logs. of Divisors.

•250000

031250

*015625

⚫003906

000244

⚫000004

⚫951243. Log. 243 = 5 x 477121 = 2·385606.
27 = 3 x 477121 = 1·431364.
729 = 6 x 477121 = 2·862728.
81 = 4 × 477121 = 1.908485.
etc.
etc.
etc.
39. The logarithms of the composite numbers are found by
the addition of the logarithms of the factors (see Art. 19). On
this principle, the following table is constructed :-

LOGARITHMS OF COMPOSITE NUMBERS.
Log. 6 = log. 2+ log. 3 = *778151.

Logarithm of 2 = Sum 301029

35. To find the Logarithm of any Prime Number.-Rule 2. Look for the tabular number nearest to the given prime number, but greater; divide the former by the latter; divide the quotient by the tabular number nearest it, but less; and so on, as before, till the last quotient coincide with some tabular number; then, the last quotient with all the divisors but the first are the tabular factors of the first quotient. Consequently, if the sum of the logarithms of these factors, which is the logarithm of the first quotient, be subtracted from the logarithm of the first dividend, the remainder will be the logarithm of the given prime number. On this principle, the following table, exhibiting another method of calculating the logarithm of 2, is constructed :

The latter logarithm of 2 is more correct than the former, owing to the difference in the mode of calculation. The logarithm of 2, calculated to ten places of decimals, is 3010299957. 36. As the prime number 5 is the quotient of 10 divided by 2, its logarithm is found on the principle that if the logarithm of the dividend be subtracted from the logarithm of the divisor, the remainder is the logarithm of the quotient (see Art. 20). Hence the reason of the following calculation is made evident :Logarithm of 10 = 1.000000

40. The integer prefixed to the decimal part of a logarithm is called its index or characteristic. Thus, in the preceding table, the logarithm of 20 is 1301030, of which 1 is the index or characteristic, and 301030 is the decimal part or mantissa. 41. From the skeleton tables and the preceding articles, it is evident (1.) that the index of the logarithm of every number between 0 and 10 is 0; the index of the logarithm of every number between 10 and 100 is 1; the index of the logarithm of every number between 100 and 1000 is 2; and so on. Hence, generally, the index of the logarithm of every integer is a number less by unity than the number of figures which it contains. The index of the logarithm of a mixed number, being determined solely by its number of figures, is, of course, not affected by the decimal.

42. (2.) The index of the logarithm of every decimal of which the highest place is tenths is -1; the index of the logarithm of every decimal of which the highest place is hundredths is 2; thousandths, -3; and so on. Hence, generally, the index of the logarithm of every decimal is a number denoting its highest place, with a negative sign attached to it. The use of this sign, which is usually written above the index, is to indicate that when the logarithm of a decimal is added, its index is to be subtracted, and when the logarithm of a decimal is subtracted, its index is to be added.

43. In tables of logarithms, only the decimal parts or mantissæ of the logarithms of the natural numbers are printed; hence, the preceding rules for supplying their indices are indispensably necessary for the purpose of calculation. To facilitate this process, however, the following table is added

Natural Numbers.

For Decimals.

Logarithms.

Natural Numbers.

Logarithms.

1.462398

Tenths

Hundredths.

Thousandths

Tenths of Thousandths

Hundredths of Thousandths.
Millionths

Teuths of Millionths

38. The logarithms of the powers of a prime number are found by multiplying its logarithm by the indices of those

44. As an additional illustration of the principles on which

Mettre fin, to put an end.

[tune.

Avoir sujet, to have reason.
Avoir tort, to be wrong.
Chercher fortune, to seek one's for-
Courir risque, to run the risk.
Demander justice, to demand justice.
Demander pardon, to beg pardon.
Donner avis,
to inform.

Faire part,

}

Entendre raillerie, to bear jokes.
Faire attention, to pay attention.
Faire bonne chère, to live well.
Faire crédit, to give credit.
Faire envie, to excite envy.
Faire honneur, to honour.
Faire horreur, to inspire horror.
Faire peur, to frighten.
Faire mention, to mention.
Faire naufrage, to suffer shipwreck.
Faire place, to make room.
Faire plaisir, to oblige.
Faire présent, to present.
Faire réflexion, to reflect.

Faire tort, to injure.

§ 83. THE

Mettre ordre, to arrange.
Par dépit, etc., through spite.
Perdre courage, to lose courage.
Porter envie, to envy.

Porter malheur, to cause misfortune.
Pour récompense, etc., as a reward.
Prendre congé, to take leave.
Prendre feu, to catch fire.
Prendre garde, to take care.
Prendre haleine, to take breath.
Prendre médecine, to take medicine.
Prendre racine, to take root.
Rendre compte, to account.
Rendre justice, to render justice.
Rendre service, to oblige.
Rendre visite, to visit.
Sans peine, without difficulty.
Sans souci, without sorrow or care.
Tenir compagnie, to accompany.
Tenir lieu, to take the place.
Tenir parole, to keep one's word.
Trouver moyen, to find means.
ADJECTIVE.

KEY TO EXERCISES IN LESSONS IN FRENCH.
EXERCISE 174 (Vol. III., page 270).

1. Ce monsieur tient-il table ouverte ? 2. Il tient un hôtel à Paris. 3. Pourquoi tenez-vous les fenêtres ouvertes ? 4. Nous les tenons ouvertes parceque nous avons trop chaud. 5. Votre ami n'a-t-il pas tenu sa parole? 6. Il a tenu sa parole; il tient toujours sa parole. 7. N'avez-vous pas dit à votre écolier de tenir la tête droite? 8. Je lui ai dit de tenir la tête droite et les yeux ouverts. 9. Pourquoi ne tenezvous pas compagnie à votre sœur? 10. Parceque j'ai promis d'aller chez mon cousin ce matin. 11. Avez-vous défendu à votre petit garçon de teuir de tels propos? 12. Je le lui ai défendu. 13. Tient-il un langage insolent ? 14. Non, Monsieur. 15. À qui vous en tiendrezvous ? 16. Je m'en tiendrai à ce que j'ai dit à M. votre père. 17. Avez-vous fait tenir cet argent à votre ami? 18. Je ne le lui ai pas encore fait tenir. 19. Le lui ferez-vous tenir demain? 20. Je le lui ferai tenir, si j'ai une occasion. 21. Pourquoi ne vous tenez-vous pas debout? 22. Parceque je suis fatigué. 23. Pensez-vous que la couleur de votre habit tienne? 24. Je crois qu'elle tiendra; elle parait trèsbonne. 25. Ne regarderez-vous pas de près aux affaires de votre frère ? 26. Je n'y regarderai pas de près. 27. Je m'en tiendrai à votre opinion. 28. Ne tenez-vous pas à votre opinion? 29. Je n'y tiens pas trop. 30. Votre médecin ne tient-il pas trop à son opinion? 31. Il y tient. 32. Cette dame vous tient-elle lieu de mère ? 33. Elle me tient lieu de mère. 34. Notre cousin nous tient lieu de père.

(1.) We have seen [§ 18, Rule (3)] that an adjective relating 35. Ce médecin ne tient pas à la vie de son malade. to two substantives of the same gender must agree with them in gender, and be put in the plural;

(2.) And Rule (4) that an adjective relating to two or more nouns of different genders, must be put in the masculine plural. (3.) When, however, nouns, united or not by the conjunction et, are somewhat synonymous; when the writer wishes actually to qualify only the last; or when the mind, more particularly occupied with the last noun, seems to forget the others; the adjective will assume the gender and number of the last noun only :

(4.) The adjectives excepté, except; passé, past; y compris, including; ci-joint, ci-inclus, annexed, inclosed; franc de port, postage free, eome under the two last rules:

Vous trouverez ci-joint la copie de la lettre que M... m'a écrite,

Le dessin de cet oiseau m'a été envoyé d'Angleterre, avec la description ci-jointe,

the letter which M. has written to

me.

The drawing of that bird came to me from England, with the descrip

tion here annexed.

EXERCISE 175 (Vol. III., page 270).

1. Have you been long at variance? 2. I have been on bad terms with him more than a month. Sir, he has been dead ten years. 3. Is your friend still alive ? 4. No, 5. Is your correspondent on his way to Paris? 6. I believe that he must have arrived. 7. Is not that young man late? 8. Yes, Sir; he never comes in time. 9. Do those engravings belong to you or to your bookseller? 10. They are mine; I have just bought them. 11. Are you not afraid of being in the way here? 12. We are on too good terms with our host to fear that. 13. Whose turn is it to fetch the books? 14. It is my turn to fetch them. 15. Does it behove you to punish him when he deserves it? 16. It is my place to punish him, for I am as a father to him. 17. Do not those houses belong to our landlord? 18. They do not belong to him. 19. They belong to our correspondent. 20. Whose letters are these ? 21. They are not mine, they are my cousin's. 22. That watch belongs to him. 23. Are you not on the eve of starting for London ? 24. We are on the eve of embarking for Cadiz. 25. We have long been in open rupture.

EXERCISE 176 (Vol. III., page 271).

1. Êtes-vous en état de le payer? 2. Je ne suis pas à même de le payer; je n'ai pas reçu mon argent. 3. Êtes-vous bien avec votre libraire? 4. Je ne suis pas bien avec lui. 5. Combien de temps y a-t-il que vous êtes brouillé avec lui? 6. Il y a plus d'un mois. 7. N'êtes-vous pas à même de satisfaire la demande de mon ami? 8. Je suis à même de la satisfaire. 9. Êtes-vous en 10. Non, Monsieur; je suis en chemin pour chemin pour Naples ? Rome. 11. Votre médicin n'est-il pas à la veille de partir pour Montpellier ? 12. Il est à la veille de partir pour Paris. 13. Est-ce que je suis de trop ici ? 14. Non, Monsieur; vous n'êtes pas de trop. 15. À qui est-ce à parler ? 16. C'est à moi à parler et à lire. 17. Est-ce à moi de lui faire des excuses? 18. C'est à votre frère de lui faire des excuses. 19. Est-ce à vous de punir cet enfant ? 20. C'est à moi de le punir, 21. Lui tenez-vous lieu de père? 22. Je lui tiens lieu de père. 23. Cet habit est-il à vous ? 24. Non, Monsieur; il n'est pas à moi; il est à mon frère. 25. Êtes-vous brouillé avec lui ? 26. Il y a deux mois que nous sommes brouillés. 27. Cette grande maison n'est-elle pas à vous ? 28. Non, Monsieur; elle n'est pas à moi; elle est à ma sœur. 29. Est-ce à votre frère de lui reprocher sa bonté ? 30. Ce n'est pas à lui de le faire. 31. A qui est-ce à aller chercher les livres? 32. C'est à moi à les aller chercher. 33. Monsieur y est-il ? 34. Non, Monsieur; Monsieur n'y est pas; mais Madame y est.

EXERCISE 177 (Vol. III., Page 317).

1. Have you not a repeater? 2. I have a gold watch, with a double case. 3. Does it go better than mine? 4. It does not go well; it loses one hour a day. 5. Has it a second hand?

6. It is a watch

You will find annexed the copy of with a second hand and a gold dial. 7. Does not your clock strike?
9. Why do not those
8. It no longer strikes, the bell is broken.
clocks agree?
10. Because one gains and the other loses. 11. Have
12 I broke it in
you not broken the main-spring of your watch?
13. Is your clock right? 14. Yes, Sir, it is right; I
winding it up.
have just had it regulated. 15. Is the striking part of this clock out
of order? 16. The striking part is out of order, and the bell is
cracked. 17. The short hand of my flat watch is broken. 18. The
pendulum of your clock is not straight. 19. How much does your
clock gain? 20. It gains five minutes a day. 21. The perfection of
a clock is not speed but regularity. 22. Does your watch stop often?
23. It stops every morning. 21. Your clock has stopped.

(5.) An adjective used adverbially, that is, modifying a verb, is of course invariable [§ 67 (3)]:En Laponie, une peau d'hermine coûte quatre ou cinq sous; la chair de cet animal sent très-mauvais,

In Lapland the skin of the ermine costs four or five sous; the flesh of this animal smells bad.

RECREATIVE SCIENCE.-XVIII.

THE ASTROMETROSCOPE-THE THAUMATROPE-THE

PEDEMASCOPE.

IN the last paper the principle of the astrometroscope, an in-
strument for producing elaborate patterns by the movement of
star-like figures, was alluded to and illustrated by Pilkington's
simple contrivance. The appa-
ratus called the astrometroscope
was devised and constructed by
Mr. Pichler, and consists of a
metallic plate, in which a number
of perforations in the form of
stars at equal distances from
each other are made, like a wall-
paper with stars on it; and when
fitted into the lantern and ex-
hibited by the oxy-hydrogen
light, white stars are apparent
on a black ground, or that part
of the disc which is not illumi-
nated. The first effect is pro-
duced by quickly moving the
perforated star-plate diagonally,
when each point of light or star-
like figure leaves a track of light,
similar to a meteor, and the effect
on the disc is that of a number
of diagonal lines moving rapidly,
every line or track of light being
perfectly distinguishable the one
from the other. The next change
is effected by imparting a general
circular motion, and if the star
figures were subjected only to

become perpendicular, and perform the same curious motion at
right angles to the former ones. In fact, it takes a quarter of
an hour to show all the figures obtainable, even when the ope-
rator makes the changes as rapidly as possible, consistent with
the separate exhibition of each pattern. Indeed, we might say
with Pope, in showing how cleverly art, in the astrometroscope,
may imitate Nature's meteors-
"All Nature is but art unknown

Fig. 1.

to thee;

All chance, direction which thou canst not see;

All discord, harmony not understood;

All partial evil, universal good."

A learned physician, Dr. Paris, appears to have been the first who specially directed attention to those more amusing effects in which we seem to be able to look through an opaque substance and see both sides of a piece of board or card upon which different designs are painted. He wrote a little book on "popular science," to which he gave the happy title of "Philosophy in Sport made Science in Earnest." It is written in the dialogue style, and is somewhat pedantic, for the worthy doctor will trot out his classics, as much as to say, "Don't think I wish to put the learning of science before the acquisition of Latin and Greek." At page 376, he thus introduces what he calls his new inven

this movement they would merge one into the other, there | tion, forgetting that from time immemorial children had twirled would appear so many circles of light and dark bands, and no other result could be obtained.

In this apparatus the movements are different, and the operator can at pleasure impart either a diagonal, circular, perpendicular, or horizontal motion, or a combination of them all. Thus a multiplication of wellknown forms can be projected on the disc, such as straight lines, then the same lines thrown into waves, next into parabolic curves, circles with star-like figures in the central portions (shown in Fig. 1), elliptical figures, or oblate spheroids.

Just as the kaleidoscope, with constant movement, may produce by reflection an endless variety of figures, so the astrometroscope, on a different principle-viz., by the formation of tracks of light left upon the vision after a variety of complicated movements of points of light-forms an endless series of line patterns. The observer is reminded of those beautiful geometrical figures obtainable by the slide-rest and overhead motion attached to the most expensive lathes.

round coins and seen both sides at the same time. But let Dr. Paris speak for himself. His preface runs thus:-"A new Optical Toy, invented by the Author, and termed the Thaumatrope-Explanation of its Prin Retentive Power of

ciple
Retina":-

Tom's holidays were now drawing to a close, and the children were summoned into the library to receive their last lesson in philosophy.

"You have lately witnessed an experiment," said Mr. Seymour, "which must have convinced you how liable the ear is to be deluded with respect to the nature and direction of sound; I shall now show you that the eye has also its sources of fallacy."

"If you proceed in this manner you will make us Cartesians," exclaimed the Vicar.

"I shall illustrate my subject by means of a toy which I have lately invented," said Mr. Seymour; "and, unless I am much mistaken, it will afford as much amusement to the elder as to the younger members of our party, although the Vicar may perhaps regard it as a more hostile instrument than even that of the wooden horse which filled unhappy Troy with an Fig. 2. armed enemy. It is a small machine," continued Mr. Seymour, "which is well calculated to furnish us with some capital puns and well-pointed epigrams."

A large book might be devoted to the illustration of the various designs obtainable from the astrometroscope; space alone must limit the drawings of these figures to one more example (Fig. 2). This pattern looks like a piece of crochetwork, the crossed loops in the circles and portions of circles in the oval figures being very curious, and only equalled by other effects, in which the lines annear like shelves or thin boards, one above the other, each shelf having so many circles upon it, and moving hal round horizontally and returning again to its former position, as if the shelves were on pivots; or the lines

"With puns!" exclaimed the horrified Vicar, who no sooner heard this appalling declaration than, like another Laocoon, he deprecated the introduction of the donum eritiale (hurtful gift) within the walls of Overton Lodge. But his hostility was soon disarmed, not by the

of deception; that everything with which they present us ought to The Cartesians maintained that the senses were the great sources be suspected as false, or at least dubious, until our reason has con firmed the report.

combined figure; on tightening the elastic string in the manner described, the card changed its axis without the slightest halt or hesitation in its rotation, and the rider was in an instant canted over the head of his charger; in a moment, however,

During this altercation, Mr. Seymour had procured the "wooden he appeared remounted; after which, by pulling the elastic

"Let us hear the account of its operation," said the Major, "which I perceive is enclosed within the box." "True," replied Mr. Seymour; "and I think you will agree that I have given a very plausible explanation of its effects."

"Plausible," muttered the Vicar -"plausible enough, no doubt; oh, the Sinon!" (Illusive is the synonym of plausible.)

Mr. Seymour then proceeded: "This toy is termed the Thaumatrope."

"Of Grecian origin!" observed the Vicar. ""Timeo Danaos et dona ferentes,' as Virgil has it."

"What is the meaning of the term?" asked Louisa.

The Vicar explained to her that it was compounded of the Greek ποτὰς θαῦμα and τρεπω, the former of which signified vonder, and the latter to turn.

"Exactly," replied Mr. Seymour, "a wonder-turner, or a toy which performs wonders by turning round. But let me proceed in the ex