in Spain. Hence arithmetical figures were brought into Christian Europe and called Arabic figures, when, in all propriety, they should have been called Hindostanee or Sanscrit characters or digits; because they were found in that country, and in the Sanscrit annals. Having traced the origin of arithmetical symbols or figures, as I believe, to their true source-in the antediluvian world, I proceed to the second part of my subject, to show the superiority of the English method of notation and numeration, over the French method; which latter, in my opinion, has been very imprudently copied into too many American works on arithmetic. 1. It is affirmed by such as have borrowed the French method of notation and numeration, that it is by far the most simple; yet, from all I can ascertain, without one single argument for the preference. This conclusion, to my mind, is the more surprising, because it really appears to have no foundation in truth. How can it be more simple than the English method? The two are precisely alike for the first nine figures: then the difference commences, in which the supposed simplicity consists. To the learner, the difficulty of comprehending the first nine figures of the numeration table, is the chief effort. He proceeds; units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions. Is it not as easy to keep on, thousands of millions, tens of thousands of millions, hundreds of thousands of millions-the plain English method-as to change the designation, and say billions, tens of billions, hundreds of billions, in the French method? To my view, the change is embarrassing and complicated. I put it to any competent judge, to say whether the English is not as simple, yes, more simple than the French method; so that the supposed simplicity of the latter, is really no other than an empty sound, or a mere figment of the imagination. 2. Let all due honor be awarded to the French nation, for important advances in science. In high numbers, in English works on mathematics or physical science, calculated according to English notation, if the French method be applied, it produces on the young mind doubt and confusion in regard to such numbers. In many instances, as I have myself witnessed, the practice has given rise to positive inconvenience and decided error. 3. A very formidable objection to the French method of notation and numeration, compared with the English, is here presented, in the designation of periods. The foundation of the French method, or, as technically it may be called the modulus, (a term derived from modulus, little measure, the base in the construction of logarithms,) is unquestionably the mille, or 1000. If the mille or 1000 be multiplied by 1000, the product makes the million, or second power of the mille or 1000. This number, however, according to strict accuracy, is misnamed; technically correct, it should have been called the bi-million, or second power of the mille or 1000-in other words, the billion; but as if determined to be inconsistent, when the mille or 1000 is their modulus, the French have raised it to the second power, or bi-million, and still have called it the mille number or million. It is certainly by involution, or the raising of powers, that they form their several periods in notation. To secure the next period after the million, which they call the billion, the mille or 1000 is multiplied into itself three times, or raised to the third power, which, in mathematical accuracy, ought to have been called the trimille or trillion; yet the inaccurate name of billion is retained. After the first two periods, the same objection lies against every period in their notation. For example, the mille or 1000 being the modulus, and the million being the second power of the mille, ought to have been called the billion; their billion, being the third power of the mille, ought to have been called the trillion; their trillion is the fourth power of the mille, and so on. Of consequence, the names of all the periods except the first two, of six figures, are technically incorrect. 4. In the English method of notation, the mille or 1000 multiplied by 1000, as with the French, makes the million, the second power of the mille; and this latter number is the modulus of the English method. A criticism might here be made on the name Million, used by the English as their modulus of notation, because it is the second power of the mille or 1000. I believe, however, the objection to this period of notation, inaccurately named, by the French plan, will not here apply. The million is arbitrarily assumed, as the modulus of the English method; as the mille or 1000 is the modulus of the French method. When the million is thus assumed, its powers, in succession, are regularly noted. The modulus is regularly involved, so as to make the name of each subsequent period mathematically and technically correct. The million multiplied by the million, makes the bi-million, or, contracted, the billion, the second power of the million; the tri-million or trillion, the third power of the million; the quadrillion, the fourth power of the million; and so on, even up to the millillion, the 1000th power of the million, or a regular series of 6006 figures entering into the English notation and numeration. Each period, then, consists of six figures, as the French does of three. The whole is vastly less wordy than the French, which, by changing names so frequently, embarrasses. The conclusion is therefore irresistible, the English method is more simple, accurate and perspicuous than the French. 5. Again, suppose it be doubted that the mille or 1000 is the modulus of the French notation, then the only alternative that remains, is to assume the English modulus or million as the French modulus; still the difficulty of incorrect names is in full force, or not at all removed. The billion of the French is formed by multiplying the mille or 1000 into the million, which will not make the bi-million, as the word indicates, that is, the second power of the million, but merely 1000 multiplied into a million. The trillion of the French, which is actually the fourth power of the mille, is only, in fact, the second power of the million, and corresponds with the correctly named billion of the English notation. The same irregularity runs through all the periods after the million, supposing the million to be the modulus. 6. To show how differently the two notations appear, and how improper it is, as I believe, to introduce the French method into English seminaries, an example or two of high numbers will suffice. According to the general conclusions of distinguished astronomers, the star Sirius is at the distance from our sun of twenty billions of miles, English notation; by the French, accurately told, it is twenty trillions. Again, Dr. Nieuwentyt has computed that there flow, in one second of time, from a burning candle 418,660,000,000000,000000,000000,000000,000000,000000, i. e. 418 septillions 660,000 sextillions of particles of light, English notation. In the French method of notation, it will make 418 tredecillions 660 duodecillions. How wide the difference between the two methods, where only 45 figures are concerned in the series! The French changing names twice as fast after the million, and yet the new names are not indicative of the power of the modulus. Who does not perceive, that American or English youth, in reading English works, of English notation, where high numbers, say from 40 to 60 figures are concerned in the series, will be utterly confused by applying the French method! Where, I ask, from the examples given, is furnished the superior elegance and simplicity of the French method? These boasted qualities of the plan vanish into empty smoke. To do justice to the French method, I insert 18 figures. Here are six periods, and the whole is 987,654,321,987,654,321-987 quadrillions, and so on. In the English notation, this series will be expressed in three periods, and the whole numeration is 987,654 billions, and so on. 7. In conclusion, I remark, that this subject demands consideration. The observations here made show its importance, however fastidious critics may demur. A hasty, yet I trust an honest examination, has been attempted, and if there be error, it shall be promptly corrected. The periods of English notation and numeration, may be made through an indefinite series or combination of figures. The highest numbers in common use, may not exceed 50 or 60 figures. Ludolf Vanceulen, in a calculation of the ratio of the diameter to the circumference of a circle, carried the series to 128 figures; which may be seen even now at Leyden, Holland, cut on the tomb-stone of that mathematician. This number, when the numeration is made, will reach the first two figures of the twenty-second period of English notation, called the unvigintillion; but by the French method, it will make two figures of the forty-third period, and be called the unquadragintillion. An example of English notation and numeration is given, pushed to 168 figures, or 28 periods, in which the million is involved to the 27th power, as the notation-name shows. That the specific names of periods may appear throughout the vast number, they are here written: Sextillions, Quintillions, Quadrillions, Trillions, Billions, 000000,000000,000000,000000,000000,000000,000000 Millions, Thousands, Tredecillions, Duodecillions, Undecillions, Decillions, Nonillions, Octillions, Sep'illions, 000000,000000,000000,000000,000000,000000,000000, 13 12 11 10 9 8 14 21 tillions, lions, 20 19 18 17 16 15 lions, Cavigintil 28 27 This whole series amounts to the immense number of 990,000 sep or septinvigintillions; or, as stated, to the 27th power of the million-a number no doubt greater than the amount of every particle of dust, all told, of ten solar systems like ours-the whole reduced to the finest particles possible! Who can comprehend the vast idea? Calculations, in some branches of physical science, approximating to this mighty number, have been made. How would such an amount look in its elegant French dress? It will be named 990 quaterquinquagintillions! Our subject stops not here. The trigintillion comprehends 31 periods, or 186 digits or figures; the centillion, 101 periods, or 606 figures; the millillion, 1001 periods, or 6006 digits or figures-high enough, yet not limited. Thus do notation and numeration, the basis of all arithmetical calculation, develop and illustrate their intrinsic importance. The excellence of the form of digits, symbols or figures in combination, shows their origin from Heaven, or more than mortal. The power of numbers, even in this feeble effort is overwhelming; human intellect fails to comprehend! Who but Omnipotence and all perfection can understand? CINCINNATI. ELIJAH SLACK. Algebra. For the Ohio Journal of Education. Ir is only within a few years that this branch of mathematics has been taught to any considerable extent in the public schools of our country. It was formerly supposed that none but students in the higher seminaries of learning could pursue this study to advantage; but it is now found that it can be taught with facility to a large class of pupils in our public schools, and there are, at this time, comparatively few common schools in which it is not receiving some attention. One cause for this change is to be found in the fact, that the character of the common schools has materially improved within a few years; and another reason is, that modern authors have so simplified the subject as to adapt it, in its elementary principles, to the comprehension of youths of ordinary mathematical talent. The subject of Quadratic Equations is highly beautiful and interesting to the student in Algebra; and the design of this article is to present to teachers and algebraic students a simple formula to determine the value of the unknown quantity in any affected quadratic equation, without the necessity of completing the square according to the rules laid down in the books. Take the equation mx2+nx=a, our formula stands thus: x= √4am+n2+n 2m This expression may be employed to advantage in equations, in which the common method of completing the square would involve troublesome fractions; e. g., 5x+11x=42, by our formula, x= V4, 5, 42+112+11 =2 or 4}, |