is a colorless gas, of somewhat acidulous taste. Not only does it not support combustion, but it furnishes one of the simplest means of extinguishing flame. A lighted taper is immediately put out by it. The choke-damp of the coal-mine, after an explosion of fire-damp, is carbonic acid. It is irrespirable. When shaken with lime-water, carbonic acid forms a white compound called Calcium carbonate, or carbonate of lime; lime-water is therefore a test for carbonic acid. The gas is so heavy that it may be poured like water from one vessel to another. For the same reason it may be collected by displacing air, as it is half again as heavy as air.

Owing to its weight, Carbonic acid is liable to collect at the place of its formation. Thus: in unused wells in which decaying leaves may be found; in coal mines after an explosion, etc. Care should be taken to explore such places before an attempt is made at cleansing, and this is easily done by the proper use of a light. It may be safely concluded that where a candle will not burn brightly, a man can not live.

In water, Carbonic acid is very soluble, especially if the pressure be increased. One measure of water dissolves one measure of the gas, and this quantity increases regularly for each additional atmosphere. By a pressure of two atmospheres water can be made to dissolve twice its measure of carbonic acid. Under a pressure of thirty-six atmospheres, carbonic acid is compressed into a liquid. All the gases have been reduced to a liquid state by low temperature and pres

sure.

Carbonic acid may be prepared by burning charcoal in oxygen, but it is most easily prepared by the action of sulphuric acid or muriatic acid upon marble. Being a compound carbonic acid must be represented by some other method than by the use of the initial letter of its name, hence chemists have adopted a system of formulas to represent all such compounds. The formula for carbonic acid is CO,. This properly interpreted means that for every atom of Carbon there are two of Oxygen. The name Carbonic dioxide is frequently applied to this gas.

Salts of Carbonic acid are called carbonates. Washing soda is sodium carbonate. Pearl-ash is potassium carbonate. Chalk, Limestone, Marble, and Shells, are varieties of calcium carbonate. White lead is lead carbonate. Malachite is copper carbonate.

Carbonic oxide is also a gas, colorless and tasteless. It contains less oxygen than carbonic acid, and may therefore be burned. When lighted, it burns with blue flame, and the oxygen of the air which is necessary to its burning changes it into carbonic acid.

Carbonic acid and Carbonic oxide are both irrespirable; the latter is, however, much more injurious than the former. Ammonia is a compound of nitrogen and hydogen. It is a colorless gas of pungent taste, and with smell as of "Hartshorn." When pure it is quite irrespirable; diluted with air, it acts as a stimulant. It does not support combustion, but is itself somewhat inflammable, burning with very pale yellow flame into water and nitrogen.

Ammonia possesses an alkaline reaction. The word alka lind was the name given to Potash, to denote that it constituted the essence of the plant that yielded it. The word is derived from the Arabic article al, "the," and kali, the name of the plant. A body has an alkaline reaction when it turns red litmus blue. The solubility of ammonia in water is remarkable. One measure of water at 15° C dis solves 783 measures of the gas. Such a solution is generally sold, and the gas may be easily prepared by heating the solution in a flask. As ammonia is lighter than air, it can be collected in a wide test-tube, by simply holding it over the neck of the flask. It is represented by NH.

Hydrogen nitrate-also called nitric acid-is a strong, fuming, volatile, corrosive liquid; very poisonous and stains

the skin yellow. It becomes yellow in the sunlight, and gives off ruddy fumes. Even when largely mixed with water, it reddens blue litmus paper, so that it possesses opposite properties to ammonia. This is called the acid reaction.

Hydrogen nitrate is made from certain nitrates. Potassium nitrate is saltpetre, so much used in making gunpowder. Barium nitrate is employed in making Green-fire, and Strontium nitrate in making Red-fire.

SULPHUR AND PHOSPHORUS.-Sulphur* is an element; a solid of a pale yellow color. Its atom compared with Hydrogen is thirty-two times as heavy. It is met with in the uncombined state, or "native," in the blue clay of Sicily, as well as generally on the coasts of the Mediterranean. It is sometimes found in transparent, yellow crystals of considerable size. When mixed with clay, the sulphur is easily separated by the aid of heat, inasmuch as it is volatile, and may be sublimed in absence of air.

In commerce Sulphur is known as Brimstone, and as Roll sulphur. Also as Flowers of Sulphur, so much used in medicine.

When sulphur is melted, from being so thin a fluid that it may be poured out like water, as is the case at first, at a higher temperature it becomes so tenacious, that the containing vessel may be turned upside down without any loss of liquid. If now the sulphur be furthur heated to its boiling point (440° C.), and then suddenly cooled by pouring it into water, it is found to be plastic and may be cast into moulds, or pulled into threads. If further heated, but still in absence of air, the sulphur volatilizes, and the vapor condenses into "Flowers of Sulphur." Sulphur melts at 115° C. It is insoluble in water. If heated either in air, or in oxygen, it oxidizes, and becomes converted into a gas possessed of most suffocating properties: this gas is called Sulphurous acid, and great care should be taken not to breathe it.

Many combinations of sulphur are met with in nature. One of its compounds, Hydrogen sulphide, is a very common emanation from sewers, and from all decomposing animal matters containing sulphur and hydrogen. Lead sulphide is the common ore of lead called Galena; Zinc sulphide is Blende; Copper and Iron sulphide is Copper-pyrites; Iron-pyrites is Iron sulphide; Silver sulphide is known as Silver-glance; Quicksilver sulphide, as Cinnabar or Vermilion.

Hydrogen sulphide, or sulphuretted hydrogen, is a compound of hydrogen and sulphur. It is a colorless gas, of most offensive odor, as of rotten eggs. It is irrespirable and poisonous. Water dissolves three volumes or measures of the gas and acquires all its properties.

Hydrogen sulphate (Sulphuric acid, or Oil of vitriol) is the most important, and generally the most powerful, of all compounds of hydrogen. It is a dense, oil-like liquid. It is much heavier than water; more than four-fifths; its relative weight is 1:84. The liquid boils at 338° C. and distils. Most organic substances char or blacken when introduced into hydrogen sulphate; this is owing to its affinity for water. When added to water, great heat results, and the mixture requires caution.

Even a drop of hydrogen sulphate in a pint of water will redden litmus paper. When added to a carbonate, carbonie acid is set free. Diluted hydrogen sulphate is employed in making Hydrogen, the zinc takes the place of the hydrogen, and forms zinc sulphate. The formula of sulphuric acid is H2SO..

There are many Sulphates in common use. Green vitriol is Iron sulphate; blue vitriol is Copper Sulphate; white vitriol is Zinc sulphate; Epsom salts is Magnesium sulphate,

* Symbol S.

Alabaster and Selenite are calcium sulphate, or lime sulphate.

Phosphorus is an element which was discovered by Brandt in 1669. The name is derived from the Greek words signifying "light-carrier," expressing one of the properties of phosphorus, viz., its luminosity in the dark. Phosphorus is never found native, but in an oxidized form it is an abundant constituent of bones, and, in smaller quantities, of all vegetables and animals, and of all fertile soils.

Phosphorus is a pale, wax-like, lustrous solid, very nearly twice as heavy as water, in which liquid it is insoluble. Its melting-point is so low that it requires great care in handling, for the moment of melting in the air is also the moment of inflammation. For purposes of experiment it should always be cut under water. Common phosphorus is so inflamable that it must be kept under water. It is generally sold in sticks, which are easily cast in moulds under warm water. Phosphorus is very poisonous. The burns which it inflicts are most dangerous. The changes which phosphorus undergoes when exposed to light, or when heated in absence of air, are very remarkable. The most important variety is the red phosphorus, which is made by heating common phosphorus for several days, in total absence of air, at a temperature of about 250°. When thus prepared it forms a red powder, insoluble in all liquids, free from poisonous properties, and only inflammable in air at a high temperature. Its atomic weight is 31.

When Phosphorus is heated in air, it burns with beautiful white light into Phosphoric acid. This substance is a snowwhite powder, possessing the most greedy attraction for water, so that in the air it rapidly flows into a liquid. When once united with water, even a white heat fails to separate it. Thus it may be melted, and then presents the appearance of ice, and is known as glacial phosphoric acid.

Phosphorus is so inflammable that we have already employed it for removing oxygen from the air. At a whiteheat carbon will take the oxygen from phosphoric acid, and thus we may prepare phosphorus.

admixture with a large volume of air. Water dissolves twice its volume of the gas, and the solution acquires the color and the properties of Chlorine. When the water is at the freezing-point, Chlorine unites with it to form a crystalline solid.

The affinity of Chlorine for hydrogen is so great that it will decompose water, or hydrogen oxide, in presence of sunlight; the result being hydrogen chloride, and oxygen. Many of the heavy metals take fire when plunged into chlorine. Dutch leaf, pounded antimony, and arsenic, at once form chlorides of the respective metals. Every metal has its chloride; nearly all chlorides are soluble in water. The two exceptions are: silver and mercurous chloride.

Chlorine is a great bleaching and disinfecting agent. This is quite intelligible. The most injurious gases, as hydrogen sulphide, and hydrogen phosphide, are instantly decomposed by chlorine; hydrogen chloride is formed, and either sulphur, or phosphorus, deposited. In sunlight, chlorine decomposes water; a portion, at least, of the oxygen is separated with the valuable properties of ozone. In cases of bleaching, chlorine removes hydrogen from the colored compounds, and thus destroys them. A chlorine atom weighs 35.5 that of H.

Hydrogen chloride* is the only compound which hydrogen forms with chlorine. It is a colorless gas, irrespirable, uninflammable, and possessed of all the properties common to acids. It fumes in air and is very easily soluble in water. At the freezing-point, water dissolves 500 times its own measure of the gas. This solution is often called hydrochloric, or muriatic acid. The gas, as well as its solution, reddens litmus. When hydrogen chloride is brought together with ammonia, white fumes of Ammonium chloride or sal-ammoniac are formed. Hydrogen chloride is not only used for making Chlorine, but also for making Carbonic acid.

Bromine is a liquid at common temperatures, of a rich red-brown color. It was discovered by Balard in sea-water, in which it occurs in very small quantities as magnesium bromide. The relative weight of bromine is three times that of water, in which liquid it is very little soluble. Vapor readily rises from bromine; its color is deep orange-red, and its vapor as injurious as, and more unpleasant than, chlorine. Chlorine separates bromine from its salts. These are called bromides. Potassium bromide is much used in medicine.

* Symbol P.

+ Symbol, Cl.

Iodine is also an element, and, like chlorine and bromine, never found native. It was discovered by Courtois, in 1811, in the ash of sea-weed. Kelp is the ash of various kinds of sea-weed, and in it is the salt of iodine, sodium iodide, from which iodine is obtained. Its symbol is I, and its atomic

Iodine resembles graphite in appearance, and is a very heavy solid, with almost metallic luster. Although but little soluble in water, to which it communicates the tint of brown sherry, it is very soluble in alcohol, and forms the Tincture of Iodine of the druggist. Iodine melts a little above the boiling-point of water, and boils at 180° C. Its vapor is of a splendid violet color, for which reason the name iodine was given. Although, when taken internally, it is virulent poison, as an outward application, especially in swellings of the glands, iodine is one of the most valuable of medicines. The salts of iodine are called iodides.

Fluorine in its elementary state is unknown, but there is every reason to believe that it is a gas. The salts of fluorine are called fluorides: fluor-spar is a fluoride of calcium. It is sometimes found massive, and of such beauti

*Formula HCl.

+ Symbol Br. Atomic weight 80. ‡ Symbol F.

ful colors, and so transparent, as to be employed for ornamental purposes. Yet its chief use is in etching glass.

Hydrogen fluoridé is employed in etching. The art was first practised at Nuremberg, and is now in general use. After careful cleaning, the glass is covered with a varnish made from wax and turpentine. The graver is then passed over the varnish, cutting through it so as to expose the glass, and sketching whatever the artist may desire. The fumes of hydrogen fluoride, when directed against this surface, attack the glass only where it is uncovered, eating, as it were, the design out of it. Atomic weight 18.

A careful study of these four elements will show how closely related they are to one another. They all unite with one atom of hydrogen, and can take the place the one of the other, atom for atom. Hence these elements, together with hydrogen, are called monad elements, from the Greek word signifying one. Chlorine, Bromine, Iodine, Fluorine, and Hydrogen are then monads.

[TO BE CONTINUED.]

READINGS ON MATHEMATICS.

II.

Geometry is the science which investigates the forms, magnitudes, positions and movements of portions of space. The etymology of the name (ge, earth, and metron, measure) suggests the probable origin of the science. As arithmetic must have originated in very early times, from the necessity 46. Origin of geof making computations of the value of primiometry. tive articles of merchandise, so the first principles of geometry must have been practically applied to the very first measurements of land. It is claimed that the science originated in Egypt, where the overflow of the Nile would be apt to obliterate the land-marks, and render their reëstablishment by systematic surveys a frequent necessity. It is even possible that the pyramids may have been constructed, in part, with a view of assisting in such surveys.

But geometrical ideas had their beginnings long antecedent to the pyramids. The rudest flint implements were constructed geometrically, and their use required the observance of laws of motion in straight lines and curves.

The principal objects of geometrical study are the proper47. Objects of ge- ties and relationships of straight and curved

ometry.

lines and their combinations-the surfaces bounded by them, and the solid forms bounded by these. Plane geometry investigates the properties of triangles and circles, and their combinations, into plane figures, bounded either by straight lines or circular arcs. Since every plane figure which is bounded by straight lines may be subdivided into triangles, the determination of the properties of the triangle involves those of all plane rectilinear figures whatever. It is a matter of the first importance, therefore, to study the triangle thoroughly, with reference to the relations of all its parts to each other and to other geometrical figures. The starting point of plane geometry is the comparison of triangles, by imagining them to be superim48. Methods. posed, the one upon the other, in such manner that certain parts of each, which are equal by hypothesis, shall exactly coincide. It is then easily proven that the remaining parts must coincide, and that the triangles are equal.

The "reductio absurdum" is a second favorite mode of demonstration adopted by geometricians. It consists in assuming the falsity of the proposition to be demonstrated and then proving the assumption to be inconsistent with either the conditions of the proposition, or with some principle previously established.

In the study of the circle and its parts, it becomes necessary to resort to a third mode of demonstra49. The Circle. tion, called the method of exhaustion. The area of a circle, unlike that of a polygon, can not be determined by subdividing it into triangles. The curvature of

the circumference renders such a subdivision impracticable. Nor can the circumference be estimated with that entire accuracy which is attainable with the periphery of the polygon. It is often incorrectly stated that a circle "may be considered" as a polygon with an infinite number of sides. This is an ad captandum statement, which, at first sight, seems to be true and sufficient, but which lacks the strict accuracy required by geometry.

the diameter.

A circle is less than any circumscribed, and greater than any inscribed polygon. By increasing the 50. Ratio of the number of the sides of the polygons, their per- circumference to ipheries will approach each other continually, until the space between them shall be less than any assignable quantity, but it can never be wholly exhausted. Consequently, the circumference of the intermediate circle can never be exactly determined by the method of exhaustion. It remains an unknown value, forever to be approximated unto, but never to be attained. One of the earliest approximations thereto was that of Archimedes, who proved that the ratio of the diameter to the circumference is nearly that of the numbers 7 and 22. It is more nearly that of 1 to 3.14159, which is the ratio commonly taken in arithmetical computations. This is so nearly correct, that it would not be practicable to detect the error by actual measurement. If a more precise result be desired, it may be obtained by extending the number of decimal places in the number representing the length of the circumference of a circle whose diameter is unity. This is the number represented by the Greek letter я, so often employed in algebraic operations relating to geometry. (19.) Its value, to thirty-five decimal places, is 3.14159265358979323846264338327950288+, which in a circle of the diameter of Neptune's orbit (5,492,542,000 miles), would be correct to within the billionth part of a hair's breadth. It would seem that no closer approximation could be desired, but, in point of fact, the decimal has been extended to 208 places, giving a result inconceivably nearer the truth, yet differing from it by an inexhaustible remainder. This slight inaccuracy, if such it may be called, is carried through all plane and solid figures bounded by curves. Nevertheless, these last may be compared with each other with perfect accuracy. Thus, Archimedes proved that the volume of a sphere is exactly two-thirds that of the circumscribing cylinder, and that the volume of a cone is one-third that of a cylinder of equal base and altitude.

One of the most beautiful of geometrical demonstrations, is that of the celebrated proposition said to 51. The Pythagohave been first taught by Pythagoras,- that rean Proposition. the area of the square described upon the hypothenuse of a right-angled triangle, is equal to the sum of the areas of the two squares described upon the two other sides.

Figure 6.

The above figure (6) suggests a simple mode of illustrating this property of right-angled triangles. The triangle is right-angled at C. The twenty-five unit squares built upon the hypothenuse A B, are equal in number to the sum of those built upon the two sides A C and B C. For a full demonstration see Loomis' Geometry, Book IV, Prop. xi.

To enumerate the endless applications of this famous proposition would require a volume. It is the basis of whole departments of mathematics, and scarcely a branch of the science is independent of it. It is of inestimable value in astronomical calculations and in surveying, navigation, architecture, etc., and it may be considered as a keystone to the whole structure of pure and mixed mathematics (62). Closely connected with plane geometry is the subject of "conic sections." If a cone be intersected by a 52. The conic sections. plane, the section may be either a circle, an ellipse, a parabola or a hyperbola. These different sections are represented in Fig. 7. The cone, whose vertex is F, and

A

B

Tigure 7. F

its base, the circle G H, is intersected by the several planes B, C, D, E and A. The curves produced by the intersections of B and C are ellipses. That produced by the intersection of D is a parabola. The section made by E is a hyperbola, and that by the base plane, A, is a circle. Although any of the above curves may be generated by other modes, their relationship with the cone has given them the name of conic sections. Their peculiar properties were well understood and treatises were written upon them long previous to the Christian era, the names of Plato, Archimedes and Appollonius of Perga, being especially distinguished among the early writers upon this subject.

The treatises of the latter are especially celebrated, and entitle him to the distinction of being classed as one of the greatest of the ancient geometers. Passing over a period of nearly nineteen centuries, during which time no great progress was made in any branch of mathematical science, we come to a new era, beginning with the invention of a peculiar and felicitous mode of applying algebraic processes to the investigation of geometrical problems.

A new branch of the science thus sprang up, under the name of analytical geometry, which differs from all other 53. Analytical branches, in that it treats almost solely of form geometry. and position, and has but little to do with magnitudes, excepting as they may assist in determining position. The peculiarities of analytical geometry may be illustrated by the geographical mode of fixing the locality of any point upon the earth's surface, by determining its latitude and longitude. Suppose that instead of the equator and the meridian of Greenwich, we substitute the two fixed lines or "axes," A B and C D, intersecting each other at

G.

Then any point, P, may be located by "referring" it to the axes A B and C D. The distances PE and P F, corresponding to the latitude and longitude of the geographer, are called the coördinates of the point P, and are designated by the letters x and y. Now, a straight line, P G, may be so located, that any point therein shall be equally distant from the two axes. Let x represent the distance of any such point from the axis CD. Then y will represent an equal distance from the axis A B. As these distances vary for the different points of the line, the letters x and y represent variable quantities. These variables, together with the coördinate axes, constitute the peculiar characteristics of analytical geometry. The expression, x=y, is the equation of the line P G in the above figure, since it is true for every point of the line. To the mathematician who is familiar with the processes of analytical geometry, the equation x=y calls up the visual conception of a straight line starting from the intersection of the axes, and proceeding therefrom upward and to the right in such direction as to bisect the angle between the axes. In other words, he sees Fig. 8. In like manner the equation, x2+y2=R2, calls up the following, (Fig. 9) which is the application of the Pythagorean The equation 54. proposition (51) to analytical geometry, in of a circle.

A

Figure 9. C

the location of a circle, whose center lies at the intersection of the axes A B and C D. From the above equation, may be deduced, by algebraic processes, every property of the circumference of the circle. There are equations, also, which represent, respectively, the ellipse, the parabola, and the hyperbola, by means of which the properties of these curves are deduced in a much more satisfactory manner, and with much less labor, than by the older

geometrical method. It is even possible to embrace in one single equation the conditions expressive of all the conic sections. Thus, the equation, y2=b2÷÷a2 (2 ax--x2), expresses not only the forms of the circle, the ellipse, the parabola and the hyperbola, but it also indicates their other properties, and those relationships between them, by which the curves insensibly approach and pass the limits of each other, the circle thus insensibly passing into the ellipse, that into the parabola, and this last into the hyperbola.

bles.

The above described method of resorting to variables to 55. Uses of Varia- express the relations of the different points of a line to two axes of reference and deducing therefrom equations corresponding to the conic sections and their derivatives, was invented.by the renowned French philosopher and mathematician, Des Cartes, and was first given to the world in the year 1637. Vieta, also a French mathematician, had previously shown the value of certain applications of algebra to geometry, but these had reference to magnitudes only, and gave no intimation that Vieta had the remotest conception of the peculiar philosophy of the ingenious method of Des Cartes. He made no use of the coördinate axes of the latter, nor of the variables,-the very devices which give to the Cartesian system its wonderful power and fertility, and which prepared the way for the discovery of the still more refined analyses of the differential calculus. It is perhaps not enough to say that analytical geometry "prepared the way" for the discovery of the calculus, it really supplied the germ from which the other grew as naturally as the tree is developed from the germinating plantlet.

56. Fluxions.

37. The calculus.

The method of fluxions of Newton, and that of the infinitesimal analysis of Leibnitz, both of which contain the essential principles of the differential calculus, were hit upon independently and almost simultaneously, by their respective authors, and the impetus which had been previously given by Des Cartes was such that had they failed to discover the calculus, some one else must have soon been driven to the discovery. The differential calculus is indeed little else than an extended application of the principles involved in the use of the variables of the Cartesian system, with the difference, however, that whilst the latter is restricted to the discussion of geometrical problems, and those of form and position especially, the former extends its researches into magnitudes and quantities, and those of all kinds whatever. Moreover, the deductions of the calculus are based upon somewhat doubtful foundations, a reproach which can not be attached to analytical geometry. It is a singular feature in the structure of this highest branch of mathematics, which transcends all others in the splendor of its achievements, that it is the only one whose reasonings are based upon foundations subject to the charge of absurdity. This blem58. Its question- ish, it is true, has been partly covered up, if able foundation. not removed, but it must be conceded that there remain, in some of the operations of the calculus, all of the original errors attached to the methods of Newton and Leibnitz. The infinitesimal calculus of the latter is founded upon the assumed principle that a quantity which is less than any assignable quantity is equal to zero, or, what is the same thing, that two quantities differing by less than any assignable quantity, are equal to each other. This principle is manifestly absurd. It is rejected, as we have already seen (50), in geometrical reasonings, and yet, it is boldly and successfully employed in the differential calculus, and, strangely enough, yields results that are undeniably true. This anomaly results from the incidental cirbring forth truth. cumstance that each error in any direction is happily counterbalanced by an equivalent error in the opposite direction; and thus, in the language of Bishop Berkeley, "error may bring forth truth, even though it can not bring

59. Error may

forth science." In order to avoid the fallacy of the infinitesimal method, Newton suggested the method of limits, the fundamental principle of which is given below in his own words [Principia, Lemma I]:

"Quantities and ratios of quantities which, in any finite term, converge to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal."

By this principle, the circle, for example, is the limit of the converging circumscribed and inscribed polygons. (50.) Although, at first sight, this principle would seem to be true and reliable, it really involves the same absurdity as that of the infinitesimal method. It was unsatisfactory to Newton himself, and his method of fluxions was accordingly based upon a third principle, which is claimed by some to be wholly free from fallacy. It is contained in Lemma II of the Principia, the language of which being unintelligible to the general reader, is omitted from this paper. Suffice to say, that the Lemma introduces the element of time, and determines the properties of curves by the 'rate at which a point momentarily advances and changes its direction.

It might be inferred from what has been said above, concerning the doubtful foundations of the calculus, that its results, after all, are only approximations to the truth. Such an inference would be erroneous. The results are, in general, correct, and accurate beyond dispute. He who would wish not to be perplexed by the paradoxical problems of this wonderful engine of mathematical power must close his eyes to its absurdities, put a check upon doubts, and accept its splendid deductions in great part upon faith.

The integral calculus is but an outgrowth of, and, in some sense, the counterpart of the differential. It 60. Integral Calincludes the latter, and, indeed, all the other culus. branches of mathematics, excepting those which are either purely descriptive or practical. It furnishes the only complete solutions of certain important problems in physics and astronomical science, and is chiefly resorted to for the determination of geometrical magnitudes, whose properties are too complicated or recondite to yield to common methods.

The next important advancement in mathematical science was the origination by Monge (near the 61. Descriptive beginning of the present century) of an im- Geometry. proved system of graphic representation of geometrical forms. This new science, which has proven to be of great utility, received the name of Descriptive Geometry. Its distinctive feature is the use of two or more coördinate planes, upon which are projected the outlines of objects, according to a method which enables the imagination to locate the objects in space, in their true forms and proportions. Its principal practical application is in the representation of engineering and architectural plans. Its study constitutes an excellent preparation for the general study of projections of all kinds, and especially for the subject of linear perspective.

All the foregoing brief outlines have reference principally to the branches of Pure Mathematics. Only the general principles of each have been touched upon, and scarcely a reference has been made to their subdivisions.

These have generally some practical application, and in so far as they are thus applied, they be- 62. Mixed Mathlong to the department of Mixed Mathematics. ematics. Plane and spherical trigonometry are chiefly employed in surveying, navigation, and astronomy. Analytical mechanics and optics are merely mathematical discussions of phenomena of solids, liquids, and gases, and of the ethereal medium whose motions are manifested to us as light, heat, electricity, and chemic force.

The realities of the number, magnitude and forms of