metic, the one treating of quantity continued, and the other of quantity dissevered. Mixed mathematics hath for subject some axioms or parts of natural philosophy, and considers quantity determined, as it is auxiliary and incident unto them; for many parts of nature can neither be discovered with sufficient subtilty, nor explained with sufficient perspicuity, nor accommodated unto practice with sufficient dexterity, without the aid and intervention of the mathematics; of which sort are perspective, music, astronomy, cosmography, architecture, engineering, and divers others. In the mathematics, I can discover no deficiency, except it be that men do not sufficiently understand the excellent use of the pure mathematics, in that they do remedy and cure many defects in the wit and faculties intellectual. For if the wit be dull they sharpen it, if too wandering they fix it, if too inherent in the sense they abstract it; so that as tennis is a game of no use in itself, but of great use in respect that it maketh a quick eye, and a body ready to put itself into all postures, so in the mathematics that use which is collateral and intervenient, is no less worthy than that which is principal and intended. And as for the mixed mathematics, I may only make this prediction, that there cannot fail to be more kinds of them as nature grows farther disclosed."

compasses, and that he made calculations. I inquired what was the meaning and use of these things, and was informed that there is a science called arithmetic. I purchased a book of arithmetic, and learned it. I was told there was another science called geometry; I bought the books, and learned geometry. By reading, I found that there were good books on these two sciences in Latin; I bought a dictionary and learned Latin. I understood also, that there were good books of the same kind in French; I bought a dictionary and learned French; and this, my lord, is what I have done. It seems to me, that we may learn everything when we know the letters of the alphabet." The duke highly pleased with this account, brought the young man out of obscurity, and provided him with an employment, which left him leisure to apply himself to the sciences. Edmund Stone, F.R.S., was the author of a new Mathematical Dictionary, published in 1726; a treatise on Fluxions and Fluents, 1730; the Elements of Euclid, 1731; and a variety of other useful works, besides papers in the Philosophical Transactions.

With the example of Stone before us, we propose to begin our Mathematical Lessons with one on Arithmetic, which will be followed by one on geometry; and so on alternately, until The admirable distinction between pure and mixed mathe- we come to the applied sciences. This will leave a fortnight's matics, which is drawn by the celebrated Lord Chancellor, in interval between each lesson; and we sincerely hope that the preceding extract, remains correct to the present day; many of our readers will avail themselves of this interval to but the wonderful verification of his own prediction, has master each lesson in succession. In order to keep up the added in an extraordinary degree to both kinds. Hence, spirit of inquiry, and to bring out the latent abilities of those under pure mathematics are now included, arithmetic, algebra, who nobly and generously aspire to be learned, we shall called by Newton universal arithmetic, logarithms, or expo- insert in each number "Queries and Problems," relating nential arithmetic, and the theory of equations, probabilities, more or less to the subjects in hand. In general, these shall &c. Also geometry, plane, solid, and analytical; trigo-be proposed one week and solved the next. It is hoped nometry, or the application of arithmetic to geometry, plane, that these also will afford both pleasure and instruction to spherical, and analytical; and the new geometry, or the our readers in general. differential and integral calculus, including the whole theory of curves and curved surfaces. Under mixed mathematics, are included mechanics, which comprehends statics and dynamics, hydrostatics, and hydrodynamics or hydraulics; also pneumatics, optics, heat, electricity, and magnetism; with astronomy, plane, nautical, and physical; the latter being sometimes denominated celestial mechanics. To these may be properly added many other mixed sciences, which perpetually call in the aid of pure mathematics; as geology, geography, geodesy, land-surveying, navigation, civil, practical, and military engineering, life assurances, steam-locomotion by sea and land, &c.

It will be the object of the "Popular Educator" to make its readers acquainted more or less with the subjects just enumerated, and their application to the present state of society; and to convey instruction in such a manner, as that any one willing to learn, may do so without the aid of a master, provided he can only read and write. That this can be done, has been often proved by the history of those who have followed" the pursuit of knowledge under difficulties." The example of Edmund Stone may be cited for the encouragement of all. He was the son of the gardener of the Duke of Argyle. At the age of eight years he was taught to read; and, at that of eighteen, he had, without assistance, made such progress in mathematical knowledge, that he could read the works of Sir Isaac Newton. As the duke was one day walking in his arden, he saw a copy of Newton's Principia lying on the grass, and called some one near him to take it back to his library. Young Stone modestly observed that the book, was his own. "Yours!" replied the duke; "do you understand geometry, Latin, Newton ?" "I know a little of them," said the young man, with an air of simplicity. The duke was surprised, and having himself a taste for the sciences, he entered into conversation with the young mathematician. He asked him several questions, and was astonished at the force, the accuracy, and the candour of his answers. "But how," said the duke, "did you come by the knowledge of all these things?" Stone replied: “Á servant taught me ten years ago to read. Does any one need to know more than the letters of the alphabet, in order to learn any thing else he wishes?" The duke's curiosity was redoubled: he sat down on a bank, and requested a detail of Proceedings. "I first learned to read," said Stone; ans were then at work upon your house; I went ne day and saw that the architect used a rule and

LESSONS IN ARITHMETIC.—No. I.

arithmeo, to count, is properly applied to the science of numThe term Arithmetic, which is derived from the Greek verb bers. To a certain extent, this science must have been coeval with the history of man. As an art, it is indispensable in daily business; and the man who is best acquainted with its practical details has always the preference in every mercantile fold-to develop its principles as a science, and to show the establishment. Our object in these lessons shall be twoapplication of its rules as an art. For this purpose, it will be necessary to begin with the first principles of Numeration and Notation, and to give such rules as will enable any one to read and write a given number correctly.

LESSONS IN FRENCH.

After the tys, or multiples of ten, by the first nine names, which are again combined by addition with the same names, so as to reach from twenty-one up to ninety-nine, a new series commences by the adoption of a new name for ten times ten, or tenty-viz., a hundred. This enables us, by the help of previous combinations, to, reach from one hundred and one up to nine hundred and ninety-nine, when a new series commences by the adoption of the new name, a thousand, for ten hundred. After this, no new name occurs till we reach a million, or a thousand times a thousand. It is true, that we have adopted from the Greek the term myriad, which signifies ten thousand, and which might properly commence a new series; but it has not been admitted into the nomenclature of our system of numeration. If the same process of analogy had been followed out, a new name ought also to have been adopted for a hundred thousand; but this has not been done, evidently for the simple reason that such high numbers were seldom in use, either in speaking or in writing. Names, indeed, would have increased so fast, and their combinations would have become so laborious to remember and to apply, in any language, that the adoption of a conventional system of signs to denote numbers, was absolutely necessary to supply the wants of mankind. Accordingly, we find that at a very early period a variety of signs or characters were invented and employed to denote numbers, and to enable men not only to express very large numbers by a few of these characters, but to make calculations of various kinds, essential both for the purposes of commerce and science.

The system of numeration adopted in our language, and explained above, proceeds on the decimal scale of numbers, in which every new name or rank is tenfold, or ten times, greater than the preceding-a system which is evidently founded on the digital structure of the human hand. So natural, indeed, is the practice of counting by the fingers, that both learned and unlearned adopt it, whenever any calculation is to be made which does not require the pen. The names of the successive ranks of numbers in their decimal order, to a certain extent, are the following:

Decimal System of Numeration.
UNITS (ones).

Tens of units.

Hundreds of units.

Thousands of units.

Tens of thousands of units.

Hundreds of thousands of units.

MILLIONS (thousands of thousands).

Tens of millions.

Hundreds of millions.

Thousands of millions.

Tens of thousands of millions.

Hundreds of thousands of millions.

BILLIONS (millions of millions).

Tens of billions.

Hundreds of billions.

Thousands of billions.

Tens of thousands of billions.

Hundreds of thousands of billions, &c.

In the preceding table, the first six ranks of names are called the first period of numbers; the next six, the second period; the six after this, the third period ; and so on. It will be observed that after the first period of units, the same names are applied, in the same order, to the second period of millions; and then, to the third period of billions. This process is continued, in our system of numeration, and the table may be extended to any length required, by applying the same names to each successive period in order, the names of these periods being as follows:

The Higher Periods.

TRILLIONS (millions of billions).
QUADRILLIONS (millions of trillions).

QUINTILLIONS (millions of quadrillions).

SEXTILLIONS (millions of quintillions).

SEPTILLIONS (millions of sex illions).

OCTILLIONS (millions of septillions).

NONILLIONS (millions of octillions), &c.

The preceding system of numeration is that adopted by all
the oldest and best English writers on arithmetic; and, up to
the last name in the higher periods above mentioned, is capa-

common system of notation. In many of our recent works on
ble of expressing a number containing sixty figures in the
arithmetic, the French system of numeration is adopted,
which differs very considerably from ours, and which has the
merit of greater simplicity to recommend it; but what it gains
in simplicity, it loses in power. Instead of dividing num
bers into periods of six ranks each, they divide them into
periods of three ranks each, the first period being called units;
the second, thousands; the third millions; and so on, as in the
following table:

French System of Numeration.
UNITS (ones).

Tens of units.

Hundreds of units.

THOUSANDS (in French, Mille).
Tens of thousands.
Hundreds of thousands.

MILLIONS (thousands of thousands).

Tens of millions.

Hundreds of millions.

BILLIONS (thousands of millions).

Tens of billions.

Hundreds of billions.

TRILLIONS (millions of millions).
Tens of trillions.

Hundreds of trillions, &c.

This table, when compared with the table of the English system of numeration, will clearly show the difference between the two systems. For example, trillions in the preceding table signifies only millions of millions; whereas in the English table it signifies millions of millions of millions. This comes of the French using the word mille for a thousand, and the word million for a thousand thousand; hence, also, the confusion arising from the similarity of the names. In consequence of the French division of the numeration table into periods of three ranks instead of six, it is plain that with the exception of one period, viz., the thousands, their system is capable of expressing a number containing only half the quantity of figures which the English system can express; and is therefore so much inferior in power.

QUESTIONS ON THE PRECEDING LESSON.

What is the origin of the term, Arithmetic, and to what science is it properly applied i

How many primitive words, as names of numbers, are to be found in almost all languages?

How are names obtained for the numbers beyond ten?
When is it necessary to invent new names for numbers?

State the combinations by addition of the primitive names of numbers in English.

State the combinations by multiplication of the same primitives. Give an idea of the manner in which the names of numbers, up to one hundred, are filled up.

What is the meaning of the name myriad ?

What is the name of the next rank after myriads, which, accord ing to analogy, would have required a new name?

What method was necessary to denote numbers to prevent the increase of names?

What are the advantages of denoting numbers by characters or signs?

What scale of numbers, or system of numeration, is adopted ir English?

What is the origin of this system?

State the different ranks of numbers in our system of numeration up to trillions.

Mention the names of the higher periods, and states how many ranks each period consists of.

Give some account of the French system of numeration. State the difference between the French and English systems and the advantages of each.

LESSONS IN FRENCH.-No. I.

By Professor LOUIS FASQUELLE, LL.D. IN commencing these French Lessons, we have thought it best to begin with a chapter exclusively devoted to the pronunciation of words, and the variations which are caused in the sounds of vowels and consonants by changes in their relative position. We shall then enter into the construction of the language, and endeavour in as plain a manner as sible to make our readers familiar with its various

and peculiarities. It will greatly facilitate their progress to 9. o nearly like o in rob. Ex. robe, robe; globe, globe; have made themselves familiar with the instructions con

11.

cachot, dungeon; haricot, bean.

bientôt, soon; suppốt, supporter.

tained in the French Lessons, reprinted from the Working | 10. ô like o in bone. Ex. dépôt, deposit; prévôt, provost; Man's Friend.* We also recommend careful attention in applying the rules and exceptions to the writing and reading of exercises. This is little more than a work of the memory, and an ordinary amount of diligence at the outset will soon enable any one not only to translate but to speak the French language with ease and comfort. First, then, as to the letters and their sounds.

THE LETTERS.

The French alphabet contains twenty-five letters :

K k

M m

u. The exact French sound of this letter is not found in
English. The position of the lips in whistling, is very
nearly the position which they should have in emitting
the French u. Ex. urne, urn; lune, moon; but, aim;
tribu, tribe; tribut, tribute; élu, elected.

12. a is the u with a prolonged sound. Ex. múre, mulberry;
dú, due; crú, growth; brûler, to burn.
13. y. See 28, y.

Examples.

amas.

14.

barre.
cas, cil.

THE DIPHTHONGS.

A vowel surmounted by an accent cannot form a dipththong with another vowel, it must be pronounced separately. Ex. obeir, to obey; deité, derly; réussite, success. Exceptions, où, where-pronounced 30; i and e accented (if e follow i) form a diphthong.

A vowel surmounted by a diaresis () follows the above rule. Ex. haï, hated; païen, pagan; maïs, maize. Exceptions; u followed by e at the end of a few words, as in ciguë, hemlock, is pronounced like û alone. 16. ai is like a in fate. Ex. j'ai, I have; je ferai, I will make; baie, bay; mai, May; balai, broɔm.

dard.

effet.

ku

ke

quadre.

P P

pay

tay

u

rixe.

yeux.
zèle.

When the diphthong ai is followed by 8, d, or t, it assumes a broader sound, resembling the French è, or ai in the English word, pair. Ex. j'avais, I had; je ferais, I should make; lait, milk; laid, ugly. 17. au nearly like oh in English. Ex. taux, rate; chaud, warm. e preceding au is blended with that diphthong without changing its sound. Ex. beau, handsome; château, castle; tableau, picture; eau, water. 18. ei nearly like a in fate. Ex. beige, serge; neige, snow; seigle, rye, reine, queen; peigne, comb.

19. eu approaches the sound of u in tub. Ex. jeu, play; lieu, place; peu, little; peur, fear; chaleur, heat. Excep tions, in eu, had: j'eus, &c., I had ; j'eusse, &c., I might have; eu is pronounced like u alone.

W, called in French double V, might be added, as many 2. ia nearly like ia in medial. Ex. il lia, he bound; il cria, foreign words which have that letter, have been adopted into the French language.

2. & like a in bar, far. Ex. âge, age; château, castle; pâte, 28. y when paste; bláme, blame; crâne, cranium.

3. e nearly like u in bud, and frequently silent at the end of polysyllables. Ex. le, the; me, me; te, thee; que, that; meuble, piece of furniture; peuple, people; rime, rime. 4. é like a in fate. Ex. été, summer; amitié, friendship; élevé, raised; épée, sword.

5. è like ai in pair. Ex. père, father; frère, brother; mère, mother; elève, pupil.

6. ê nearly like e in there. Ex. rêve, dream; extrême, extreme; crême, cream; crêpe, crape; forêt, forest.

7. i nearly like i in pin. Ex. midi, mid-day; ici, here; fini, finished; credit, credit.

you;

sound in English.

French

of the

sounds.

Examples

cou, neck.

Il salua, he saluted; il remua, he moved; il contribua, he contributed.

Il salue, he salutes; il remue, he moves; il contribue, he contributes.

produit, produce; conduite, conduct; lui, him; bruit, noise; il réduit, he reduces. duo, duet,

initial, when coming between two consonants, or when forming a syllable of itself, has the sound of the French i. Ex, style, style; type, type; yeux, eyes; Ypres, Ypres; y, there; between two vowels y has the power of two i's, one of which forms a diphthong with the preceding, and the other with the following vowel ; the syllabic division taking place between the i's. Ex. moyen, means; essayer, to try; nettoyer, to clean; citoyen, citizen; abbaye, abbey; these words are pronounced as if they were written mon-ien, essai-ier, nettoi-ier, citoi-ien, abbai-ie. The words, pays, country, paysage, landscape; paysan, peasant, are pronounced pé-is, pé-isage, péi-san.

THE NASAL SOUNDS.

29. The combination of the vowels with the consonant m or n, produces what the French call le son nasal, the nasal sound.

The o of the other letters has 30. When the consonant m or n is doubled, or is immedi + se in rose.

⚫ kse. # This accent indicates the suppression of the letters after the vowel on which it is placed, thus: fête, tête, bête, were formerly written, feste, teste, beste, the s was not sounded, but gave to the preceding vowel that prolonged sound now represented by the circumflex accent.

ately followed by a vowel the nasal sound does not

• The words ennui, ennuyer, emmener, enivrer, enorgueillir, form excep tions to this rule The first syllable of ennui, ennuyer, emmener is nasal enivrer, enorgueillir are pronounced en-nicirer, en-orgueillir.