Sidebilder
PDF
ePub

32. By means of these three skeleton tables, and the principles already explained, the logarithms of all natural numbers may be found to any extent required, within certain limits as to the number of decimal figures.

33. 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 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 :

[blocks in formation]

34. 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 tablet, exhibiting another method of calculating the logarithm

of 2, is constructed :-
:-

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]
[merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

37. The logarithms of the powers of a prime number are. found by multiplying its logarithm by the indices of those powers (see Art. 21). On this principle, the following tablets are constructed:

[blocks in formation]

the addition of the logarithms of the factors (see Art. 18). On 38. The logarithms of the composite numbers are found by this principle the following table is constructed:

Logarithms of Composite Numbers.

[blocks in formation]

2.00000 = 1.07722

[ocr errors]

1.07461 = 1.00243

[ocr errors]

1.00225 = 1.00018
1·00014

...

1·00004

1·00004
1.00000

...

⚫333333 031250 ⚫000977 ⚫000061 000015

Log. 18

log. 2 + log. 3 = 778151
log. 2+ log. 6 1.079181
log. 3 + log. 61.255273

[blocks in formation]

...

[blocks in formation]
[blocks in formation]

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 10 places of decimals, is 3010299957. 35. 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. 19): Hence the reason of the following calculation is made evident : Logarithm of 10 = 1.000000 Logarithm of 2 =

[blocks in formation]
[blocks in formation]

39. The integer prefixed to the decimal part of a logarithm table, the logarithm of 20 is 1-301030, of which 1 is the index is called its index or characteristic. Thus, in the preceding

or characteristic, and 301030 is the decimal part.

40. From the Skeleton tables and the preceding articles, it is evident, First, that the index of the logarithm of every number of which the highest place is units, is 0; the index of the logarithm of every number of which the highest place is tens, is 1; the index of the logarithm of every number of which the highest place is hundreds, is 2; thousands, 3; and so on. Hence generally, the index of the logarithm of every integer is a number less by unity than that which denotes its highest place. The index of the logarithm of a mixed number being determined solely by its highest integer place, is, of course, not affected by the decimal,

41. Secondly. The index of the logarithm of every decimal | avert crime, accident from ignorance, and death from neglect, it is of which the highest place is tenths, is-1; the index of the plain that this qualification would give an impulse to education far logarithm of every decimal of which the highest place is surpassing any that it has yet received. 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 the 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.

42. In TABLE of Logarithms, only the decimal parts 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.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][ocr errors][ocr errors][ocr errors][ocr errors]

May every reader of the P. E. do his best to get up a petition to Parliament in his own neighbourhood for the introduction of such a clause. This movement will be productive of the greatest welfare of this nation and conducive to the everlasting bliss of man. J. H.

ANSWERS TO CORRESPONDENTS.

cadence," without the jingling sound of like endings." But, although J. M. C. (Plymouth): Blank verse is quite capable of "rhythmical the greatest poets, as Homer, Shakespeare, and Milton wrote in blank verse, it does not follow that "blank verse is the best of all poetry;" for "blank verse" may not be poetry at all, as we have had too frequent occasion to notice among our correspondents. Now, once for all, let our opinions on this subject be duly considered: we think that lines with like endings, as he phrases it, may hinder and retard poetical ideas with some, and assist them with others. Milton wrote his immortal poem, "Paradise Lost," in blank verse, but it would not have injured his ideas in the slightest if he had made the lines jingle with like endings; Pope wrote his translation of "Homer's Iliad," a work far superior to the original, in heroic rhyme-that is, he made the lines jingle with like endings, and yet it would not have been one whit more poetical had it been written in bank verse; for, as we have said before, Pope wrote in numbers, for the numbers came." It is true that poeta nascitur, non fit;" but it is also true that cultivation does a great

[ocr errors]

deal.

[ocr errors]

Thousands

Tens of Thousands Hundreds of Thousands

[merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

SIR,-As the second reading of the new Reform Bill is postponed until the 27th of April, I think that the readers of the P. E. should make an endeavour to get a clause inserted in the bill for the purpose of making learning, intelligence, and education, qualifications for voting, as well as money. The great objection to universal suffrage is, that the unlearned and ignorant possess not the understanding to judge who is a fit and proper person to represent them in Parliament. But, I ask, why should the wise and the intelligent be disqualified for want of money, while they have an enlightened mind and understanding to know who is a fit representative, and possess sufficient judgment to discern who is the best legislator ?

I think that a money qualification may include as many ignorant persons, in proportion, as universal suffrage. The money qualification may be the reward of industry; but may not the most industrious still be forgotten? May not that man who has a large family to bring up, be still more industrious than the old bachelor who has £50 in the savings' bank? A man who is bringing up and supporting subjects for her Majesty the Queen, subjects who may one day bravely defend Old England from a foreign enemy, protect her Majesty on her throne, and keep our Parliament at its duty, should, if he be wise and intelligent, be allowed to exercise the franchise without any other qualification.

The education qualification would be the best education-bill which the Parliament could pass; for then the nation would be self-educated-fathers would educate their sons, and their sons would educate themselves: and as our prisons and our assizes, our towns and our villages cry aloud for education in order to

LITERARY NOTICES.

THE HISTORICAL EDUCATOR.

The First Volume of this Important Work, containing Ancient Voyages and Travels, with Popular Histories of Greece and America, etc. etc., is just published, price 6s, neat cloth.

CASSELL'S LATIN DICTIONARY, BY J. R. BEARD, D.D.—The publication of this Dictionary has commenced, and will be completed in about Twenty-six Numbers, THREEPENCE each, or in Monthly Parts, ONE SHILLING each. Nine Numbers, as well as the First Two Parts, are now ready.

CASSELL'S FRENCH AND ENGLISH DICTIONARY.-The FRENCH and ENGLISH portion of this important Dictionary is now completed, and may be had, price 4s, or strongly bound, 53.-The ENGLISH and FRENCH portion is in the course of publication, and will be completed in about Twelve Numbers, THREE PENCE each. The entire Dictionary, forming one handsome volume, will be ready in a few days, price 98.6d.

CASSELL'S GERMAN PRONOUNCING DICTIONARY.-The GERMANENGLISH Portion of this Dictionary is now ready, price 5s. in stiff covers, or 5s. 6d. strong cloth.-The ENGLISH-GERMAN Portion will be completed as quickly as possible, in Numbers, THREE PENCE each; and the entire Volume, strongly bound, at 9s., will shortly be issued.

CASSELL'S LESSONS IN GERMAN. Parts I. and II.-These Lessons have been ackowledged by those who have studied them to be the easiest introduction to the German Tongue which has ever been published in the English Language. Price 28. each, in paper covers, or 2s. 6d. cloth.-The Two Parts bound together, price 4s. 6d.

A KEY TO THE EXERCISES IN CASSELL'S LESSONS IN GERMAN. In the Press.

CASSELL'S ECLECTIC GERMAN READER: Containing choice selections from the best German Authors, in Prose and Verse, with a complete Lessons in German," Parts I. and II., to which it is intended to serve as a Dictionary of all the Words employed, and copious references to "Cassell's Supplement. Price 2s. in paper covers, or 2s. 6d. bound in cloth.

CASSELL'S LESSONS IN GERMAN PRONUNCIATION: consisting of easy Extracts from German Writers, with interlinear directions for the Pronun ciation of every Word, and a Dictionary explaining the meaning of each. By means of these directions, a person knowing nothing of German previously, may at once pronounce the language so as to be easily understood by a native. Price is. in stiff covers, or 1s. 6d. neatly bound.

CASSELL'S SHILLING EDITION OF FIRST LESSONS IN LATIN. By Professors E. A. ANDREWS and S. STODDARD. Revised and Corrected. Price 1s. paper covers, orals. 6d. neat cloth.

CASSELL'S LATIN GRAMMAR. For the use of Schools and Colleges. By Professors E. A. ANDREWS and S. STODDARD. Revised and Corrected. Price 3s. 6d. in cloth boards.

CASSELL'S LESSONS IN LATIN. These Lessons have been pronounced unrivalled by thousands of students. Many who have stadied Latir. from other grammars and on other systems, and have completely failed, have acquired more real knowledge of the Latin Tongue from these Lessons in six months, than they have acquired in as many years by the means hitherto dopted. Price 2s. 6d. In paper covers, or 3s. in cloth.

A KEY TO CASSELL'S LESSONS IN LATIN. Containing Translations of all the Exercises. Price 1s. in paper covers, or 1s. 6d.

1s. 6d. cloth, consists of a LATIN READER, adapted to Cassell's Fir.t CASSELL'S CLASSICAL LIBRARY.-The First Volume of this Work, price Lessons in Latin."-Volume II. comprises LATIN EXERCISES, price 25. neat cloth.-Volume III. contains THE ACTS OF THE APOSTLES in the Original Greek, with copious Notes and a Lexicon, price 2s. 6d, neat cloth.

LESSONS IN CHEMISTRY.-No. XXIX.

Two very important precipitating agents for lead out of its solution yet remain to be discussed: they are oxalic acid or its soluble combinations, and sulphurous acid either free or combined.

The completeness with which oxalic acid, or, still better perhaps, oxalate of ammonia, precipitates lead from its solutions admits of ready demonstration. As respects the case of acetate of lead, let the pupil add oxalic acid in solution or oxalate of ammonia, until no further precipitate is determined. This condition having been arrived at, let the solution be filtered and the colourless filtrate be tested with hydrosulphuric acid solution. Under these circumstances, not the slightest indication of blackness will be visible, a fact which proves that the oxalic acid, or oxalate of ammonia, are complete precipitants of lead out of its acetic acid combination.

Proceeding with our examination of oxalate of lead, it admits of demonstration that the substance is exceedingly insoluble in water, even under the operation of boiling. In point of fact, oxalic acid and the soluble oxalates are remarkably efficient and delicate tests for lead in solution.

Extraction of Metallic Lead from Oxalate of Lead.-If oxalate of lead be mixed with carbonate of soda and powdered charcoal (black flux is perhaps better) and heated on a piece of charcoal with the blowpipe-flame, metallic lead results. It is impossible, however, to conduct the operation in such a manner that a portion of lead oxide shall not also be developed, 80 great is the tendency of metallic lead to combine with oxygen when heated to a sufficient extent in contact with atmospheric air. Oxalate of lead is easily reducible when treated as described, but it may also be stated that all lead precipitates under similar treatment afford parallel results with various amounts of difficulty. Perhaps the sulphate of lead is more difficult of reduction than any other lead compound. Sulphurous Acid as a Precipitant for Lead-The suffocating gas generated when brimstone or sulphur is burned in atmospheric air or oxygen, and termed sulphurous acid gas, is a complete precipitant for lead out of its solutions, and is extensively used for that purpose in the operation of refining sugar by sub

acetate of lead.

For the purpose of illustrating this action, prepare a solution of acetate of lead, strength immaterial, and lighting, immediately above the surface of this solution contained in a testglass, a brimstone match, blow the gaseous result of combustion (sulphurous acid gas) down upon the surface of the acetate solution. These directions being followed, a copious white precipitate will immediately appear; the white substance being sulphate of lead. The plan we have followed for rendering evident the effect of sulphurous acid on solutions of lead, is necessarily different from the plan followed in practice, whether in the laboratory or the larger scale of commercial manufacture. The sulphurous acid requires not merely to be brought into contact with the surface of a liquid, but to be passed through it in a stream. On the large manufacturing scale, the sulphurous acid generated by the combustion of sulphur in atmospheric air is forced through the liquid to be deprived of lead by means of a pump. In the laboratory, pressure can be more readily obtained by generating the acid in a flask, a process which no longer admits of burning sulphur as the source of sulphurous acid. We must employ for this purpose oil of vitriol, and generate our gaseous result by heating that liquid in contact with charcoal, or mercury or copper; in general terms charcoal is the most convenient.

Into a Florence flask, pour about a table-spoonful of wood charcoal broken into small pieces, but not powdered, to this add about a table-spoonfull of oil of vitriol, adapt a cork and bent tube as represented in the accompanying diagram, fig. 26, dip the free end of the tube to the bottom of a test-glass containing acetate of lead solution, and apply heat.

These conditions being complied with, copious volumes of two gases, sulphurous acid and carbonic acid, are evolved; permeate the liquor and throw down sulphate of lead. As to the carbonic acid it does no harm, for so long as sulphurous acid passes, no carbonate of lead is found. By proceeding thus, the operator will discover that every trace of lead is removed from the solution, which, if exposed for some days to a warm atmosphere, in order to get rid of lingering sulphurous

VOL. V.

[merged small][merged small][graphic]

Before tracing the application of sulphurous acid gas to the sugar refining process, it will be necessary to examine the effect of the acetates of lead generally (there are more than one; we have been dealing with the mono or neutral acetate) on certain organic solutions. For this purpose we will take a solution of muscovado or common yellow sugar, about the strength of one of sugar to six of water (both by weight). Solution having been effected by means of heat and agitation, add to it portions of solution of acetate of lead, or, still better, the tris-acetate, commonly known in druggists' shops as Goulard's extract, until most of the colouring matter has become deposited. I say most, because the latter portions of colouring matter cling to the sugar solution with great obstinacy, and cannot be gotten rid of without the expenditure of more acetate of lead than is from motives of economy desirable. You will observe that on now filtering the sugar solution thus treated, the filtrate will be almost devoid of colour, hence the acetate, or rather the tris-acetate of lead would be the very best substance, could it be safely used, to be employed for the purification of yellow sugar. It is quite evident, however, that the employment of this material must remain out of the question until the discovery of some inexpensive, readily-applied and unfailing plan of extracting superabundant lead has been devised. All the ordinary precipitants for lead had been tried without avail, until Dr. Scoffern bethought himself of sulphurous acid, which answers completely. The process has now been in extensive operation for more than six years, and no untoward accident has occurred from the use of the poisonous agent.

This is a convenient opportunity for the introduction of some remarks on chemical nomenclature, having special reference to the compounds of oxide of lead with acetic acid, and of very general application in other cases. You will remember that I denominated the Goulard's extract employed in our sugar refining experiment by the name of tris-acetate or sub-acetate of lead; let us therefore investigate the meaning of those terms. Tris is evidently a prefix signifying three of something; sub is a prefix which means an under or diminished quantity of something. We will proceed to investigate the exact meaning of these terms. Beginning with sugar of lead, this is the neutral acetate; it is composed of one equivalent of base (oxide of lead) combined with one equivalent of acetic acid. Now the chemico-algebraical notation or symbol for lead is Pb (contraction for Plumbum," and the symbol for oxygen is O, consequently oxide of lead (protoxide) being composed of one equivalent of lead plus one of oxygen, its symbol is Pb O.

Otherwise called sub-acetate, or Goulard's extract. 108

[ocr errors]

Again, A is the contracted symbol for acetic acid; therefore PbO+A or Pb O, A represents neutral or mono-acetate of lead. But it so happens that there are other acetates of lead, in number somewhat doubtful, perhaps five or six, all of which contain an excess of base (oxide of lead, or PbO). Indefinitely therefore they may be generalized as sub-acetates of lead; but if we desire to be precise, and indicate what kind of sub-acetate of lead any particular one may be, we must have recourse to numeral prefixes, and chemists have long agreed to indicate all excess of base in any salt by prefixes of Greek origin. Thus mono-acetate of lead remains neutral acetate; di acetate, the particular subsalt generated by combination of two of base with one of acid; tris-acetate, the particular subsalt generated by combination of three of base with one of acid, and so on for the rest. Furthermore, it should be remarked, that, in addition to the general or indefinite word sub-salt, there is another very much employed in modern chemical treatises: I mean basic salt; that is to say, a salt in which the base predominates.

And this is man:-Oh! ask of him,
The gifted and forgiven,-
While o'er his vision, drear and dim,
The wrecks of time are driven;
If pride or passion in their power,
Can chain the time, or charm the hour,

Or stand in place of heaven?

He bends the brow, he bows the knee,"Creator, Father! none but thee!"

ON PHYSICS, OR NATURAL PHILOSOPHY.
No. XXX.
CALORIC.

Continued from page 36.)

THE following table will be found of great utility in convertthe number of degrees on any of the three different scales into the corresponding number of degrees on either of the other scales, viz. Fahrenheit's, Reaumur's, and the Centigrade.

This table will be extremely useful to Students of Chemistry and Natural Philosophy, especially in reading the accounts of French and other continental discoveries in these sciences. We shall have frequent occasion to refer to it ourselves, and to make our Students refer to it in future Lessons. Experimental Chemists will do well to purchase this number of the P.E., were it for the table only, as it will save their making calculations, and prevent their committing mistakes in converting the degrees from one scale into another.

In our next lesson we shall enter upon the important processing of cupellation, or separation of lead from silver, and indeed gold by aid of fire. It is an operation of great beauty and of particular interest just at this time, when so many persons are devoting their attention to the study of the precious metals. The operation of cupellation will require the possession of some bone ashes. If the student desire to enter very far into the practice of the art, he will also require several pieces of apparatus, to be described in our next. The mere theory of the operation, however, admits of being taught by the aid of a blowpipe operation. The preparation of bone ash is a matter of no difficulty. Into a clear fire put pieces of bone, and there let them remain until every thing capable of being burned away has been removed-in other words, until they become quite white. Remove them then from the fire, and when cold powder them.

MAN.

(From the New York Evening Post.)
THE human mind,-that lofty thing!
The palace and the throne,
Where reason sits a sceptred king,

And breathes his judgment tone.
Oh! who with silent step shall trace
The borders of that haunted place,
Nor in his weakness own
That mystery and marvel bind
That lofty thing,-the human mind!
The human heart,-that restless thing!
The tempter and the tried;
The joyous, yet the suffering,-

The source of pain and pride;
The gorgeous thronged, the desolate,
The seat of love, the lair of hate,-
Self-stung, self-deified!

Yet do we bless thee as thou art,
Thou restless thing,-the human heart!
The human soul,-that startling thing!
Mysterious and sublime!

The angel sleeping on the wing

Worn by the scoffs of time,

The beautiful, the veiled, the bound,
The earth-enslaved, the glory-crowned,
The stricken in its prime!
From heaven in tears to earth it stole,
That startling thing,—the human soul!

The composition of acetic acid is 4 equivalents of carbon + 3 of hydrogen+ 3 of oxygen; hence its full symbolic notation is C4 H3 O3; but this being somewhat inconvenient, chemists usually represent it by its initial letter A with a dash over it, thus A; and here it may be remarked that a dash over a letter signifies the body to be of the organic kingdom, i. e. either of animal or vegetable origin, whereas a dash under an initial is a reduplication of quantity. Thus A means acetic acid, and A 2 of acetic acid.

In explanation of the following table, we observe that it is divided into three columns, headed Degrees of Fahrenheit, Degrees of the Centigrade, and Degrees of Reaumur, respectively. The table commences with the number of the degrees marked at the boiling point on each scale, and gradually decreases by 1 degree, and in some cases by parts of a degree, to -42°.25 Fahrenheit, -41.25 Centigrade, and -33° Reaumur. Thus, suppose that when the temperature of a body is 200° Fahrenheit, and you wish to know what it is on the Centigrade scale, and on Reaumur's; by turning to the number 200 in the column headed Degrees of Fahrenheit, you find in the horizontal line with 200, and in the other two columns to the right, the numbers 93.33 and 74.66, which indicate that the corresponding degrees of the given temperature are 93°.33 on the Centigrade scale, and 74° 66 on Reaumur's.

Again suppose that when the temperature of a body is 70° Centigrade, and you wish to know what it is on Fahrenheit's and Reaumur's scale; by turning to the number 70 in the column headed Degrees of the Centigrade, you find in the horizontal line with 70, and in the other two columns, one to the left and one to the right, the numbers 158 and 56, which indicate that the corresponding degrees of the given temperature are 158° on Fahrenheit's scale, and 56° on Reaumur's.

Lastly, suppose that when the temperature of a body is 59° Reaumur, and you wish to know what it is on the Centigrade scale and on Fahrenheit's; by turning to the number 59 in the column headed Degrees of Reaumur, you find in the horizontal line with 59, and in the other two columns to the left, the numbers 73.75 and 164.75, which indicate that the corresponding degrees of the given temperatures are 739-75 on the Centigrade scale, and 1640.75 on Fahrenheit's.

When the proposed number of degrees on Fahrenheit's thermometer exceed the limits of the table, that is, are above the boiling point, then the rules given in our last Lesson must be employed, in order to convert them into the corresponding number on the other two scales. Thus, suppose that the temperature of a body, as indicated by Fahrenheit's thermometer, was 600°, and it was required to find the corresponding number of degrees on the Centigrade and Reaumur's, we have the following calculations :—

[blocks in formation]
[blocks in formation]

Fahrenheit Centigrade. Reaumur. Fahrenheit. Centigrade. Reau ur Fahrenheit Centigrade Reaumur. Fahrenheit. Centigrade. Reaumur,

[blocks in formation]
« ForrigeFortsett »