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bonic acid, inasmuch as, under the circumstances most favorable to its production, as in respiration, it is never produced in sufficient quantity to allow of its recognition by its familiar odor of hartshorn. Nor can it ever accumulate like carbonic acid, as its very easy solubility in water occasions its constant removal from the air by every passing shower or fall of dew.

Ammonia is a compound of two elements, nitrogen and hydrogen; from it plants derive their nitrogen, without which they would be useless as food for animals, whose flesh contains abundance of nitrogen. Though surrounded by air, which contains 79 per cent. of nitrogen when freed from moisture, all plants would languish for want of it, were it not for the ammonia also present. The plant can make no use of nitrogen in its simple form; it can only ob tain it from some compound containing nitrogen. Hence the value of manures is for the most part in proportion to the amount of ammonia which they contain, or afford. How admirable, therefore, is the provision that decay and putrefaction should thus minister to the life and vigorous growth of fresh vegetation!

There is yet a third form in which nitrogen is sometimes contained in the air. In rain-water, after thunderstorms, Nitric Acid can nearly always be discovered. It has been proved by experiment, that the mere passage of sparks from the electrical machine through the air is sufficient to cause some of the nitrogen to unite with oxygen and with water, to produce nitric acid or hydrogen nitrate, as it is generally called. Indeed it was in this manner that the compound nature of nitric acid was first shown by Cavendish in 1785. As a manure, hydrogen nitrate diluted with much water, is as valuable for its nitrogen as is ammonia.

We have already learned that the atmosphere is a mixture of oxygen and nitrogen, together with more or less of the vapor of water, a very small quantity of carbonic acid, and a mere trace of ammonia and of nitric acid. But, inasmuch as the atmosphere surrounds the earth, everything that is volatile may find its way into the air, and form an occasional constituent. Whenever vegetable matter decays in presence of much moisture, Marsh-gas is always generated. It is a compound of carbon and hydrogen, and is largely a constituent of coal-gas.



[Addison is one of the brightest ornaments of English literature. He excelled both in prose and verse. His essays are regarded as the finest models of that style of writing in the language. Among his smaller poems are a few of the nature of hymns, which have found their way into the hymn-book of nearly every Christian church. The following "dream" is not untrue to nature, and points a moral. It is written in the pure, perspicuous English characteristic of the author. Addison was born 1672, died 1719.]

It is a celebrated thought of Socrates, that if all the misfortunes of mankind were cast into a public stock, in order to be equally distributed among the whole species, those who now think themselves the most unhappy, would prefer the share they are already possessed of, before that which would fall to them by such a division. Horace has carried this thought a great deal further, (Sat. i. 1. 1, ver. 1,) which implies, that the hardships or misfortunes we lie under are more easy to us than those of any other person would be, in case we could change conditions with him.

As I was ruminating upon these two remarks, and seated in my elbow chair, I insensibly fell asleep; when on a sudden methought there was a proclamation made by Jupiter, that every mortal should bring in his griefs and calamities, and throw them together in a heap. There was a large plain appointed for this purpose. I took my stand in the center of it, and saw, with a great deal of pleasure, the whole human species marching one after another, and throwing down their several loads, which immediately grew up into a prodigious mountain, that seemed to rise above the clouds.

There was a certain lady of a thin airy shape, who was very active in this solemnity. She carried a magnifying glass in one of her hands, and was clothed in a loose flowing robe, embroidered with several figures of fiends and spectres, that discovered themselves in a thousand chimerical shapes, as her garments hovered in the wind. There was something wild and distracted in her looks. Her name was Fancy. She led up every mortal to the appointed place, after having very officiously assisted him in making up his pack, and laying it upon his shoulders. My heart melted within me to see my fellow-creatures groaning under their respective burdens, and to consider that prodigious bulk of human calamities which lay before me.

There were, however, several persons who gave me great diversion. Upon this occasion, I observed one bringing in a fardel very carefully concealed under an old embroidered cloak, which, upon his throwing it into the heap, I discovered to be poverty. Another, after a great deal of puffing, threw down his luggage, which, upon examining, I found to be his wife.

Among the most curious occasional constituents of the air, is a peculiar modification of oxygen, which is known by the name Ozone. Although possessed of very different properties, it is only oxygen under different physical aspects. It was discovered by Schoenbein in 1840, and was called ozone on account of its marked odor. Those who are acquainted with the smell produced when an electrical machine is put in action, will know what ozone is like, There were multitudes of lovers, saddled with very when they are told that the peculiar smell is due to ozone. whimsical burdens, composed of darts and flames; but, Although ozone is never present in the air in anything like what was very odd, though they sighed as if their hearts sufficient quantity to be observable by its odor, yet it is would break under these bundles of calamities, they could often largely produced during storms; some compounds not persuade themselves to cast it into the heap when they containing oxygen are probably decomposed, and the oxy- came up to it; but after a few vain efforts, shook their gen set free has the properties of ozone. It is a very pow-heads, and marched away as heavy laden as they came. erful disinfectant, owing to its oxidizing properties. As it saw multitudes of old women throw down their wrinkles, is perfectly insoluble in water, the air which passes over and several young ones, who stripped themselves of a seas does not lose its ozone. When it meets with putrestawny skin. There were very great heaps of red noses, cent matter, ozone oxidizes it, enters into composition with large lips, and rusty teeth. The truth of it is, I was surthe hydrogen of the putrescent matter and disappears. In prised to see the greatest part of the mountain made up of large cities the air is also always more or less contaminated bodily deformities. Observing one advancing toward the by the various gases evolved from putrefying or decaying heap with a larger cargo than ordinary upon his back, I animal and vegetable matters, from the burning of coal, and found, upon his near approach, that it was only a natural from the numerous chemical processes carried on in such hump, which he disposed of with great joy of heart among places. this collection of human miseries. There were likewise [TO BE CONTINUED.] distempers of all sorts; though I could not but observe, that


there were many more imaginary than real. One little packet I could not but take notice of, which was a complication of all the diseases incident to human nature, and was in the hand of a great many fine people: this was called the spleen. But what most of all surprised me, was a remark I made, that there was not a single vice or folly thrown into the whole heap; at which I was very much astonished, having concluded within myself that every one would take this opportunity of getting rid of his passions, prejudices, and frailties.

I took notice in particular of a very profligate fellow, who, I did not question, came loaded with his crimes, but upon searching into his bundle, I found that, instead of throwing his guilt from him, he had only laid down his memory. He was followed by another worthless rogue, who flung away his modesty instead of his ignorance.

When the whole race of mankind had thus cast their burdens, the phantom which had been so busy on this occasion, seeing me an idle spectator of what passed, approached toward me. I grew uneasy at her presence, when of a sudden she held her magnifying glass full before my eyes. I no sooner saw my face in it, but was startled at the shortness of it, which now appeared to me in its utmost aggravation. The immoderate breadth of the features made me very much out of humor with my own countenance, upon which I threw it from me like a mask. It happened very luckily that one who stood by me had just before thrown down his visage, which it seems was too long for him. It was indeed extended to a most shameful length; I believe the very chin was, modestly speaking, as long as my whole face. We had both of us an opportunity of mending ourselves; and all the contributions being now brought in, every man was at liberty to exchange his misfortunes for those of another person.

I saw with unspeakable pleasure the whole species thus delivered from its sorrows; though at the same time as we stood round the heap, and surveyed the several materials of which it was composed, there was scarce a mortal in this vast multitude who did not discover what he thought pleasures and blessings of life, and wondered how the owners of them ever came to look upon them as burdens and griev


As we were regarding very attentively this confusion of miseries, this chaos of calamity, Jupiter issued out a second proclamation, that every one was now at liberty to exchange his affliction, and to return to his habitation with any such bundle as should be allotted to him.

Upon this Fancy began again to bestir herself, and par⚫elling out the whole heap with incredible activity, recommended to every one his particular packet. The hurry and confusion at this time was not to be expressed. Some observations which I made upon the occasion, I shall communicate to the public. A venerable gray-headed man, who had laid down the colic, and who, I found, wanted an heir to his estate, snatched up an undutiful son, who had been thrown into the heap by his angry father. The graceless youth, in less than a quarter of an hour, pulled the old gentleman by the beard, and like to have knocked his brains out; so that, meeting the true father, who came toward him with a fit of the gripes, he begged him to take his son again, and give him back his colic; but they were incapable either of them to recede from the choice they had made. A poor galley-slave, who had thrown down his chains, took up the gout instead, but made such wry faces, that one might easily perceive he was no great gainer by the bargain. It was pleasant enough to see the several exchanges that were made, for sickness against poverty, hunger against want of appetite, and care against pain.

The female world were very busy among themselves in bartering for features: one was trucking a lock of gray hairs

for a carbuncle; another was making over a short waist for a pair of round shoulders; and a third cheapening a bad face for a lost reputation: but on all these occasions there was not one of them who did not think the new blemish, as soon as she got it into her possession, much more disagreeable than the old one. I made the same observation on every other misfortune or calamity which every one in the assembly brought upon himself in lieu of what he had parted with; whether it be that all the evils which befall us are in some measure suited and proportioned to our strength, or that every evil becomes more supportable by our being accustomed to it, I shall not determine.

I could not from my heart forbear pitying the poor humpbacked gentleman mentioned before, who went off a very well-shaped person with a stone in his bladder; nor the fine gentleman who had struck up this bargain with him, that limped through a whole assembly of ladies, who used to admire him, with a pair of shoulders peeping over his head.

I must not omit my own particular adventure. My friend with a long visage had no sooner taken upon him my short face, but he made such a grotesque figure in it, that as I looked upon him I could not forbear laughing at myself, insomuch that I put my own face out of countenance. The poor gentleman was so sensible of the ridicule, that I found he was ashamed of what he had done; on the other side, I found that I myself had no great reason to triumph, for as I went to touch my forehead, I missed the place, and clapped my finger upon my upper lip. Besides, as my nose was exceedingly prominent, I gave it two or three unlucky knocks, as I was playing my hand about my face, and aiming at some other part of it. I saw two other gentlemen by me, who were in the same ridiculous circumstances. These had made a foolish swap between a couple of thick bandy legs and two long trapsticks that had no calves to them. One of these looked like a man walking upon stilts, and was so lifted up into the air, above his ordinary height, that his head turned round with it; while the other made such awkward circles, as he attempted to walk, that he scarcely knew how to move forward upon his new supporters. Observing him to be a pleasant kind of fellow, I stuck my cane in the ground, and told him I would lay him a bottle of wine that he did not march up to it on a line that I drew for him in a quarter of an hour.

The heap was at last distributed among the two sexes, who made a most piteous sight as they wandered up and down by the pressure of their several burdens. The whole plain was filled with murmurs and complaints, groans and lamentations. Jupiter at length taking compassion on the poor mortals, ordered them a second time to lay down their loads, with a design to give every one his own again. They discharged themselves with a great deal of pleasure; after which, the phantom who had led them into such gross delusions, was commanded to disappear. There was sent in her stead a goddess of a quite different figure; her motions were steady and composed, and her aspect serious but cheerful. She every now and then cast her eyes toward heaven, and fixed them upon Jupiter. Her name was Patience. She had no sooner placed herself by the mount of sorrows, but, what I thought very remarkable, the whole heap sunk to such a degree, that it did not appear a third part so big as it was before. She afterwards returned every man his own proper calamity, and teaching him how to bear it in the most commodious manner, he marched off with it contentedly, being very well pleased that he had not been left to his own choice as to the kind of evils which fell to his lot.

Besides the several pieces of morality to be drawn out of this vision, I learned from it never to repine at my own misfortunes, or to envy the happiness of another, since it is impossible for any man to form a right judgment of his neigh-

bor's sufferings; for which reason also I have determined never to think too lightly of another's complaints, but to regard the sorrows of my fellow-creatures with sentiments of humanity and compassion.


In undertaking to present the subject of mathematics in the form of "readings," the writer is aware that he must encounter peculiar difficulties. The common notion that the science is difficult and abstruse is well-founded, and, doubtless, any attempt to evade the severe labor and long continued, patient application required to master it must prove futile. Let it be understood, then, that these "readings" are intended neither to supply the place of such labor, nor to suggest any short process for transforming the unsophisticated reader into a mathematician. They must be considered merely as an introductory prelude to that closer study which the reader shall hereafter bestow upon a subject, proficiency in which can not be acquired without persistent mental effort.

No one need hope to invest this subject with the charms of a romance or the attractions of a novel. It has nothing in common with the sensibilities of human nature. It is wholly and purely intellectual, and although it frequents the profoundest depths of the realms of the imagination, it goes forth in pursuit of no ideal of goodness or loveliness. It is in search only of those mysteries of number, quantity, and form which dwell amid the infinities of time, space, and force.

Yet there is an indescribable charm attending the pursuit of mathematical truth, due in part to the greatness and dignity of the quest. It widens the horizon of the imagination and extends the intellectual vision far beyond the bounds of things visible and tangible. It places in the hand of the philosopher the measuring line wherewith to measure the universe, and the balance in which to weigh the omnipotencies of matter and force.

vince of matics.

The true province of mathematics, in its higher depart1. The true pro- ments, at least, is to take part with the physical sciences as an indispensable aid to research, and a means of attaining to the greatest possible degree of accuracy of knowledge concerning the numberless phenomena of nature.

It is amid the common affairs of every day life, however, that the masses of mankind are more clearly 2. Its practical application in com- brought under the reign of mathematical law. mon life. Every industrial, commercial, or other pursuit engaged in by any person, requires him to take some note of quantities, of time, space, or force, or of numbers of things.

We divide time into periods, space into fragmentary portions, and force into its component forces. We unite frag. ments into masses, and group individuals of species into pairs, or dozens, or hundreds, or myriads, and almost every moment of life we light upon something to be measured, or counted, or estimated, or meet with some form to be observed or amount of force to be considered. In common business transactions endless calculations become necessary with reference to money, produce, weights, dimensions and every imaginable element of exchangeable value.

All such calculations, however simple or trivial, are mathematical processes, and thus every human being is practically a mathematician, whether he be versed in the refined rules and processes of algebra and the calculus, or keeps his accounts by notches upon a stick, or scores upon the wall.

The primitive custom of keeping count by fives and tens, 3. Primitive math-obviously originated from the use of the fingers, and the invention of the ten so-called


Arabic characters or figures was doubtless suggested by the same habit. The system of notation in common use throughout the civilized world is therefore due to the anatomical structure of the human body. It might have been based upon the aber eight, had a four fingered race originated it, or, indeed, upon any other number, down to two, as was once suggested by a celebrated mathematician. It is thus but one of many possible arithmetical systems that might have subserved the same purposes equally well. For some reasons the number eight would be preferable to ten, but from habit and familiarity with the denary system exclusively, we experience difficulty in conceiving the possibility of abolishing any of the ten Arabic characters or the places occupied by them. The great excellency of the system is not due to the number of characters, but to the highly ingenious device of assigning to each character both an intrin❘sic and a local value—the latter depending upon its position with reference to other figures. The local value varies by multiples of ten, any figure representing ten times the value which it would express if removed one place to the right. Had any other number than ten been used as a base, the increase in value of each figure would have been in multiples of that base. Thus, on a system of eights, instead of tens, the numbers 10, 100, 1000, would indicate, respectively, 8, 64, and 512.


The study of the philosophy of numbers has its attractions, and much might be said concerning them, but further consideration of this subject must be reserved for a future paper. The common uses of numbers, involved in every day's occurrences, are the first beginnings of mathematical science, and shall first claim attention.

These papers being written especially for those who may have paid but limited attention to this branch of study, their mission is to present the theme in such manner as shall convey to the reader some general notion of the scope and methods of each important branch without the requirement of minute or critical study of them.

Mathematics is the science which investigates the laws and relationships of real or ideal forms, positions, magnitudes, numbers, and quantities. It is divided 4. Definition of into several distinct branches, the more im- Mathematics. portant being those of arithmetic, algebra, geometry, analytical geometry, and calculus. The other branches are either subdivisions or extensions of these. Thus, trigonometry, and its application to surveying, navigation and astronomy are extended branches of geometry. Fluxions, now nearly obsolete, is but a more cumbersome form of differential calculus, differing from it only in its notation, and "conic sections" is a purely geometrical branch relating to properties of the cones produced by the intersections of a plane with the surface of a cone.

Arithmetic is that branch of mathematics which occupies the first place in a course of mathematical 5. Nature of study. It is the oldest branch, with the ex- Arithmetic. ception of geometry.

Moreover, it stands in the front rank in importance, both on account of its universal applicability to the common concerns of business, and its indispensability as a foundation for the whole science of quantity-no other branch thereof being wholly independent of its aid.

It deals principally, if not wholly, with number, and hence its fittest definition is comprised in four words: "the science of number." The name Arithmetic is derived from the Greek word signifying number, and the science includes all that may be considered in every numerical process or statement relating to quantities of time, space, or force, and to numbers of discrete things of every kind.

Since every variety of quantity has its own kind of units and specific modes of subdivision thereof, 6. Arithmetical there are, of necessity, almost as many arith- Rules.

metical rules and processes as there are kinds of quantity to be considered. Some of these are quite complicated, and, ofttimes, to the beginner, difficult and puzzling.

The various intricacies of compound numbers, of square and cube root, of percentage, etc., may well be styled ditficult to those who may have happened to contract a dislike for numbers, and even the brightest student may sometimes find them puzzling.

Nevertheless, the actual fundamental operations of arithmetic are simple enough, and require but little 7. The four fundamental operations effort for one to become familiar with them. of arithmetic. They consist simply in adding, subtracting, multiplying, and dividing. These are the four only processes to which numbers require to be subjected, and they serve not only for calculations pertaining to arithmetic, but for those needed in every branch of mathematics.

But whilst the fundamental operations of arithmetic are thus few and simple, there are many definitions and principles to be regarded in performing them. Some arithmeticians, for example, enumerate seven or eight principles concerned in the rules for multiplication, and a like number for those of division. Some of these refer to the properties of numbers, which may be considered as 8. Properties of numbers. prime or composite, concrete or abstract, integral or fractional, simple or compound, odd or even, square, cubic, triangular, etc.

Others, again, refer to the kinds of units. Thus, there

are units of time, of weight, of extent of surface, of length, of capacity, of monetary value, etc., and of discrete things in endless variety.

In some calculations several kinds of quantity are involved, and their respective properties and relationships must be concurrently borne in mind. Thus, in problems concerning longitude and time, regard must be had to the angular motion of the earth on its axis and in its orbit, measured by signs, degrees, minutes, and seconds; also, to the lapse of time, which is estimated in seconds, minutes, hours, days, and years, and often also to geographical position, fixed in part by linear measurements.

higher branches

Again, the simple operations of arithmetic are frequently 9. Arithmetic de- Complicated by reason of their intimate rependent on the lationship with problems belonging to the of mathematics. higher branches of mathematics. When two equal numbers are multiplied together, their product is called a square number, from the circumstance that its units may be symmetrically arranged in the form of a square. Thus the number "25" is a square number, because if its units be represented by objects of any kind they may be arranged as shown in Fig. 1.

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The relationship between arithmetic and the higher branches is further shown by the fact that all rules relating to the mensuration of surfaces and solids, and to involution, evolution and ratio, have either a geometrical or algebraic basis.

Figure 2.

For example, the rules for the extraction of the square root are derived from the mathematical law-which may be demonstrated by either geometry or algebra- 10. The square that "if a quantity be divided into any two root of numbers. parts, the sum of their second powers, together with twice their product, is equal to the second power of the whole quantity. A geometrical representation of the above law is given in Fig. 3, in which the line A B represents the

quantity which is divided into two parts at the point C. The more accurate statement of the geometrical law is that "if a straight line be divided into two parts, the sum of the squares described upon the parts, together with twice the rectangle contained by them, is equal to the square described

upon the whole line."

Figure 3.

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The full elucidation of such rules as those for square and cube root, therefore, belongs rather to the higher branches of mathematics. They are referred to here simply as illustrations of those arithmetical processes which are based upon algebra and geometry, and which render some portions of arithmetic somewhat difficult to understand. The explanation of them to beginners is much facilitated by the use of blocks and other models, hereafter to be described. (45). The process of repeatedly multiplying any quantity by the number of its units is called involution. The successive products are called powers, and the original numbers roots or factors. If the root be contained but twice as a factor, the power is usually called its square; if thrice, its cube, and, in general, the nth power contains the root n times as a factor. (25).

12. Involution.

Any succession of numbers increasing or diminishing by any special law, is called a series: if increasing, an ascending, and if decreasing, a descending series. If the series be formed by successive applications of a common difference, it is called an arithmetical progression; but, if formed by repeated multiplications by a common ratio, the progression is geometrical. It is evident that a power is a geometrical progression whose ratio is equal to the first term. In every progression there are five quantities considered, the first and last terms, the number and sum of the terms, and the ratio, any three of which being given, the other two may be computed by rules derived from algebraic formulae. These rules embrace forty cases, many of which furnish problems difficult or impossible to solve without the aid of algebra.

The astonishingly rapid increase in the value of the terms, or successive products of a geometrical progression is well

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To give standing room to this enormous number would require the surface of a globe millions of times larger than the sun and all the planets put together.

Besides the rules for the above specified general operations, the arithmetic of numbers includes 15. Special promany special processes suited for special kinds cesses.

of business, and for the solution of particular scientific problems.

Mensuration, for example, embraces almost as many rules as there are varieties of form to be considered-such as polygons, circles, and ellipses, among surface forms; and cubes, prisms, pyramids, polyhedrons, spheres, spheroids, ellipsoids, cones, cylinders, etc., among solids.

Alligation treats of mixtures of substances of different values or prices. It is a branch of percentage, which includes rules for interest, discount, banking, and kindred computations.

Finally, some treatises on arithmetic devote special departments to problems in mechanics and other branches of physics, and, on the other hand, most branches of the physical sciences have their own peculiar arithmetical rules relating to the particular subjects discussed by them.

He who has thoroughly mastered the fundamental rules of arithmetic may enter upon the study of al- 16. When to begebra, where he will find use for all that he gin the study of Algebra. has previously learned concerning number and quantity, and where he will be introduced to a new system of notation employing many additional symbols, and requiring a severer study of methods of operation which are by far more complex than those of arithmetic.

Algebra is an arithmetical branch of mathematics, in which the quantities and numbers are, in part, 17. Definition of represented by letters of the alphabet, and their Algebra. relations are indicated by special characters and devices. Its operations, in some respects, are identical with those of the arithmetic of numbers, but they are of much more extended and general application, owing to the superior system of notation employed, and to the advantageous use of various symbols, and ingenious expedients, by means of which many of its operations are greatly abridged.

The especial algebraic symbols of quantity being letters of the alphabet, to which no fixed or definite values are assigned, algebra is aptly defined as that branch of mathematics which treats of indeterminate quantities.

This is its distinguishing feature as compared with arith


metic. A solution of a strictly algebraic prob- 18. Distinguishing lem, is a solution, not of one particular case features of Algeonly, but of all possible cases falling within the conditions of the problem. Many of the rules of arithmetic are derived wholly or in part from these general solutions of algebraic problems, and their infallibility could not well be proven by any purely arithmetical process. The utmost that such process could prove is the application of the rule to each specified case.

Good examples illustrative of this point are furnished in the rules for the square and cube root, and, in the development of the law, that "the product of the sum of two num

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