definition and the prominence of its position. It can never be mistaken.

The sandstones of the lower carboniferous reach a noted development in Ireland, and may be reckoned hardly second in importance to the arenaceous accumulations of the preceding period, the old red sandstones. While the deposits of coal are so noted that they not only have no equal, but not an approach to a rival, these three important features of this period together make the carboniferous the most celebrated of all the systems.

In England the system is developed in many places, but the main axis is a broad rectangular patch which stretches from the border of Scotland to the town of Derby; it occupies the whole breadth of the island, as far south as Whitehaven on one side, and Hartlepool on the other. In both cases the line then leaves the coast, still preserving a more or less perpendicular direction, till on one side it reaches the neighbourhood of Crewe, and on the other the town of Nottingham; then a line joining these places and passing through Derby will form the southern boundary of the axis. A glance at a geological map will show all the places where the members of the system appear, far more accurately than we can describe them.

The most convenient division of the strata which lie between the old red sandstone beneath and the new red sandstone above, seems to be

1. Lower Coal Measures.

2. Mountain Limestone. 3. Millstone Grit.

4. Upper or True Coal Measures.

1. The lower coal measures include all the strata which appear between the old red sandstone and the mountain limestone; they are but scantily developed in England, but appear in full force in the south of Ireland, and also in Scotland. They bear a subdivision into

1. Carboniferous Slates. 2. Yellow Sandstones. The yellow sandstones, sometimes named after the Irish geologist, Sir R. Griffiths, Griffiths' sandstones, rest unconformably on the old red on the southwest coast of Ireland, and are again and again exhibited to the very north of the island.

These yellow sandstones vary in their thickness from 400 to 2,000 feet, and it is but right to state that their position is not yet finally established; there is much reason for believing them to be co-ordinates of the Dura Den sandstones, a group certainly Devonian, which are developed in Fifeshire, and contain fossils of the Holoptychius and Pterichthys, which are eminently creatures of that period. Yet, until this is firmly established, we prefer to leave them where they are generally inserted, as the lower members of the carboniferous group; and indeed there is much direct reasoning in favour of this position; for although in them, as in the rest of the sandstones, fossils are few and far between, yet those which do occur possess strong carboniferous relations: for example, specimens of the Lepidodendron, Calamitis, Stigmaria, and Cyclopteris have been discovered, and, as we shall find, these are the main species which the woods of the carboniferous world contained. These yellow sandstones are found in England in Gloucestershire, and in South Wales.

Carboniferous Slates.-The upper division of the lower coal measures is divided by Professor Jukes into

3. Coomhola Grit Series.

2. Dark Grey and Black Slates and Shales.

1. Lower Limestone Shale with calcareous bands.

The Coomhola Grits are yellow sandstones into which are inserted grey and black shales; they are found in the immediate neighbourhood of Bantry Bay, and are but scantily fossiliferous.

The Dark Grey and Black Slates and Shales are very similar to the series which lies beneath them, and are yet a formation which was deposited under somewhat altered circumstances; for they contain no slates, which marks an absence of deep water where the argillaceous sediment quietly reached the bottom.

The Lower Limestone Shale is, in reality, the transition bed between the lower coal measures and the mountain limestone. In this bed the calcareous bands begin to appear, and, on the other hand, in the great deposit immediately above it all sandstones disappear, and the calcareous matter gains the predominance.

This series begins to bear evidence of the fossiliferous character of the mountain limestone, for in the calcareous

bands which traverse it, an abundance of marine fossils are discovered.

The lower coal measures in Scotland, though not so typically or so fully developed, yet reach a considerable thickness in the districts of Fife and Lothian; they have none of the slaty character of their Irish relations, but are mainly white sandstones, interstratified with bands of dark bituminous shales. As in Ireland, these shales owe their dark colour to the presence of that vegetable matter which, in the upper parts of the series, accumulated to form seams of coal.

In Ireland it never exists in a separate form, though in Scotland it appears in thin seams of coal. The iron which tinged the deposits of the Devonian period with red, seems to have increased in quantity so as to form thin bands of ironstone, which, higher up in the system, form rich deposits.

The fossils which this series contains, proclaim that the beds have an estuary origin, partly marine and partly fresh-water; and to this fact may be ascribed those peculiarities which have been judged sufficient to rank these beds as a distinct group.

All corals which are exclusively marine existences are conspicuously absent, although in the mountain limestone immediately above they occupy a prominent position. The bands of limestones, such as those at Kingsbarns, Fife, which sometimes are three feet in thickness, are composed entirely of shells of bivalves of the mussel tribe; and when we know the habits of these mollusks make them love an estuary existence, we have another fact confirmatory of the fresh-water neighbourhood in which the lower carboniferous was formed. Moreover, fragments of land-shells, and some remains of small reptiles of the frog tribe, show the proximity of land and the influx of rivers.

2. Mountain or Carboniferous Limestone.-This deposit is always dark-coloured and semi-crystalline; from its hardness it forms bluff and bold escarpments, which may be seen in many parts of Yorkshire, Derbyshire, Westmoreland, Fife, and Ireland. It has received its name "mountain" from the fact that it frequently occupies the higher grounds of the carboniferous period.

It occasionally is found in masses several hundred feet thick, which are here and there traversed by bands of shale. These masses are split up into rectangular blocks by the joints and backs to which we alluded in a foregoing lesson; but often

this limestone is found in beds alternating with seams of coal and shales and sandstones; yet let it be in what position it may, the most crude geologist will at once pronounce its name. It forms a distinct zone among the rocks; a land-mark in the geological landscape. It is an excellent building-stone, and may be seen in Ireland forming walls built with singular ingenuity of fragments of the rock fitted almost like irregular mosaic work.

In the iron districts it is used in the smelting furnaces to rid the ore of its silica, with which the limestone forms a fusible slag, and so melts out; or again, the agriculturist burns it in the lime-kiln, and uses the lime on his land; then it is frequently quarried for ornamental work, since it takes a good

69), Archemidopora (Fig. 70), Amplexus (Fig. 71), and Syringopora (Fig. 72). These, as the reader will notice, are widely different in their shape, but there are many immediate forms. One of these, the Amplexus (Fig. 71), is a cup coral. Milne-Edwards showed that in the paleozoic period (which, if reference be made to the summary of geological strata in Lesson XI., will be found to contain all the life which is exhibited in a fossil state to the top of the Permian period), the corals bore a simple and striking relation to all those of the same kind which were produced in the neozoic period.

A longitudinal and a horizontal section of the paleozoic and neozoic cup corals are shown in Figs. 73 a, b, and 74 a, b. In Fig. 73 a, b, are represented those sections of the paleozoic

So fine-grained a limestone is an admirable material for preserving the records of the past life of the globe. The fossils of this member of the carboniferous system are numerous and perfect: since sometimes the whole of the rook is one mass of corals and encrinites, showing its true marine character, we shall enumerate these first.

The corals are the cells which a soft polyp built round itself; the creature has the power of extracting from the waters of the sea the carbonate of lime which they contain. These cells were arrayed round a common axis, in a certain symmetrical form, or sometimes they were built one on the other in defiance of any peculiar order, such as the reef corals of our

Beas.

Some of the most prominent of these we have illustrated; namely, Lithostrontion basaltiform (Fig. 68), Aulopora (Fig.

coral, and upon comparing them with Fig. 74 a, b, at once it will be seen that the cross sections of each show a different arrangement of the lamella, or the walls of division; for while in Fig. 736 they are in fours and multiples of 4, in Fig. 74 b the governing number is 6. While the other sections show a difference, the palæozoic has transversal plates, the neozoic has none. This alteration in the form of the structure of the coral at the change of this period of life is peculiar.

The Encrinites are the distinguishing feature of the mountain limestone; they are, in fact, star-fish on stems, only the rays of the star-fish are greatly increased. They present the intermediate step between the free and fixed Crinoidea. They have all passed away save two species, which are occasionally found, but are very rare, the surviving members of a vast race. Such a great development did the encrinites reach in the carboniferous period, that we find masses of limestone wholly composed of the stems of the stone-lilies, as they are rather poetically called. This stone is often seen polished in

mantelpieces; when exposed to the air, the softer parts, which are immediately between the stems, weather away, leaving the surface of the rock as if the fossils had been stuck on to it, as in the illustration of encrinital limestone (Fig. 75). The heads of the animals which the stem supported were a corona of tentacules, with a mouth at the centre. These arm-like feelers waved about in the waters, bringing into their mouth the prey which they enclosed. As specimens of these encrinites, we give the Cyathocrinites planus (Fig. 76) and the Woodocrinus (Fig. 77).

Fossils of the sea-urchins are also found, but these paleozoic echini had more plates than are found in specimens of their existing representatives, and their detached spines are of common

occurrence.

The Mollusca appear in great force. The trilobites still struggle for existence, and are represented by two members of the race-Phillipsia (Fig. 78) and Griffithides. The crustaceans have other representatives, such as the Eurypterus (Fig. 79). The fish of the preceding age, of course, have not become extinct, and many perfect fossils remain; moreover, species variously formed are continually met with, which indicate the existence of other fish armed with defensive weapons. These sharp fin-spines were formidable, as may be judged from the appearance of the Pleuracanthus (Fig. 80) and the Ctenacanthus (Fig. 81).

There is also evidence of the presence in the carboniferous seas of fish, the roofs of whose mouths were armed with palatal plates. We should have been at a loss to have even conjectured the use of these smooth, black, and oblong hard pieces of enamel, had it not been that in Australia there is a shark-the Port Jackson shark-whose mouth is paved with these very plates. The creature lives on crustaceans, and to break, for instance, the shell of a crab, sharp and pointed teeth would be useless; so that he is furnished with plates by which he crushes and breaks the hard coats of his food, and so lives. These fossils were vulgarly called fossil leeches. Two of these palatal plates of the Placoid sharks we figure, namely, Cochliodus contortus (Fig. 82) and Psammodus porosus (Fig. 83). The chief fossils of the period are also represented in the illustration in the preceding page.

Some of the above fossils are very numerous: for example, the Productus sometimes all but forms the whole mass of the rock, which is then called a Productus limestone.

The chambered shells, as represented here by the Euomphalus, the Goniatites, and the Orthoceras, have many peculiarities which command notice.

The Bellerophon has no chambers, it is like the living Argonaut; the genus is not found after this epoch.

The Euomphalus is chambered, but there is no connection between the chambers, so that as the animal grows it builds up its shell behind it, throwing a wall or septum (Latin, septum, a partition) from side to side.

The Nautilus tribe do the same, only they have a pipe which communicates with each of the chambers: in the nautilus this siphuncle is in the centre, whereas in the ammonite tribe it is marginal, or runs round the outside edge of the shell. The creature is a cephalopod, its arms are out of its head; as it moves on the bottom of the sea seeking its food, if it dragged its delicate shell after it, it would be bruised and damaged; hence it passes, by means of its siphuncle, either water or air into the empty chambers, and so makes its shell of the same specific gravity as the water around it, and so it just swims on the surface of the water.

The Orthoceras (Fig. 84) is a nautilus unrolled, having a siphuncle passing down the centre of the cells.

The Goniatites (Fig. 85), on the other hand, is of the Ammonite type; the siphuncle is marginal.

Some of the mountain limestones are very beautiful. The Terebratula bastata (Fig. 86) frequently exhibits the original coloured stripes which ornamented the living shell.

The Aviculopecten (Fig. 87) and the Pleurotomaria (Fig. 88) also exhibit the same phenomenon, though in not so marked a degree.

The presence of these colours indicates that the sea in which they lived was not more than fifty fathoms deep.

Other notable fossils of the period are-Productus (Fig. 89), Spirifer (Fig. 90), Murchisonia (Fig. 91), Euomphalus (Fig. 92), and Bellerophon (Fig. 93).

ELECTRICITY.—IX.

ATMOSPHERIC ELECTRICITY-FRANKLIN'S KITE EXPERIMENT -CROSSE'S APPARATUS-LIGHTNING CONDUCTORS.

WE have now to notice some of the main effects of Atmospheric or Aerial Electricity. These are very interesting and very important, as it seems highly probable that by inquiring into them we shall ultimately understand much more of the phenomena of the weather than we do at present.

The many points of resemblance between lightning and elec tricity were noticed by early experimentalists. The flash of the electric spark resembles faintly a flash of lightning. A loud sound accompanies each, and both have the power of setting on fire inflammable bodies. Lightning destroys animal life, rends bad or insufficient conductors, and is attracted by the pointed tops of trees or buildings; and all these effects may be produced on a small scale by the agency of electricity. We easily see, then, the great resemblance. It was left, however, for the celebrated Franklin to prove their identity, and, by drawing down the lightning from the clouds, to render his name for ever famous in the annals of electrical science.

About the commencement of the year 1752 he began to investigate the subject, and formed the idea of erecting, on a convenient building, a tall insulated conductor, terminating in a point, by which he expected to be able to collect the electricity from passing clouds. He was unable just at the time to carry out his purpose; he had, however, given a full expla nation of his ideas in a letter to a friend, and acting upon this a French electrician succeeded in obtaining sparks from an apparatus he had prepared in accordance with these plans. Franklin was now tired of waiting for the erection of a building suitable for his purpose, and the idea occurred to him of trying his experiment with a kite. He accordingly made a framework of two thin laths crossing one another, and fixed the four corners of a silk handkerchief to their extremities. A tail was affixed in the usual way, and a loop for the string to be fastened to. In order to collect the electricity he fixed a pointed wire to the upper end of one of the laths, and connected it with the string of the kite. The other end of the string he fastened to the ring of a door-key, and fixed to this a piece of silk ribbon, so that he might hold it without the electricity escaping, and could draw sparks from the key.

Thus equipped he went out one stormy day, accompanied only by his son, and having raised the kite on a common near Philadelphia, awaited with almost breathless anxiety the result. For some time no effects whatever were seen; there was not the slightest appearance of any electrical disturbance; after a while, however, he observed some of the fibres of the cord bristling up; and on applying his knuckle to the key, to his intense delight he drew from it a spark. Soon afterwards a heavy shower of rain fell, and the cord having thus become wetted, its conducting power was much increased, and a rapid succession of sparks was given off from the key, by which a Leyden jar was charged, and other experiments performed which proved beyond doubt the absolute identity of lightning and electricity.

Franklin was thus the real discoverer of this fact, though some have attempted to claim this honour for the French philosopher we have referred to, who was named Dalibard, on the ground that he erected the first apparatus by which the experiment was performed.

Since this time the experiment has been repeated over and over again in a great variety of ways. Frequently a fine wire is woven into the string, and greatly increases its conducting power. When this is not done it should be moistened with salt and water. Captive balloons have likewise been employed to draw down the electric fluid. The utmost caution is, however, requisite in conducting these experiments, as fatal accidents have arisen in carrying them out. Not very long after the date of Franklin's experiment, Professor Richmann had erected an apparatus at St. Petersburg for collecting in this way the atmospheric electricity, the wires being brought into his laboratory. One day he was explaining it to a friend, just as a storm was coming on, and all at once a loud report was heard, and the electricity issued almost in the form of a ball of fire, killing him on the spot, and striking his friend senseless to the ground: many of the things in the room were at the same time shaken or destroyed.

If, therefore, any of our readers should be inclined to repeat the experiment, every precaution must be taken. A very good plan is to wind the cord upon a roller, contained in a metal case, supported on stout glass rods. A glass handle should be fixed to a multiplying wheel, so as to pay out or coil up the cord, and a large ball, connected with the ground by means of a stout wire or chain, should be placed within an inch or an inch and a half of the metal case. If then the power becomes so great as to be unsafe, the electricity will dart to this ball, and by it to the ground.

In connection with this it must be remembered that the sparks thus drawn from a kite have far greater power than those from an ordinary electrical machine. Sparks two or three inches long may be taken from the latter without any inconvenience, while one only half an inch long from the kite will often be as powerful as a shock from a Leyden jar; the power, however, does not always depend upon the length of the spark.

For making frequent observations upon the electrical condition of the atmosphere, it is needful to have something more permanent and manageable than a kite. A tall pole erected on the top of a house or lofty building, or in some elevated position, answers well. On the upper extremity of this there is fixed a thick glass rod, surmounted by a number of points, or a large metal fork, so as to attract the electricity.

The lower part of the metal should be fitted with a kind of cap, so as to keep the glass rod as dry as possible, and thus preserve the insulation in damp weather. A wire is then attached to the metal, and being carefully insulated is brought down to the wall of the laboratory, into which it enters through a glass tube. It is necessary, however, to fix a large ball in connection with the ground, within about two or three inches of the wire, before it enters the room, so that in case of a flash of lightning striking it the current may escape in this way. A similar ball and rod should also be placed inside the room, so arranged that the distance between it and the knob in which the wire terminates may be adjusted to any required distance. In this way the current may be allowed to escape quietly to the ground when it is not required. A peal of bells may also be attached, so as at once to call attention to the passing of a highly charged cloud.

The most complete apparatus for the collection of atmospheric electricity was that fitted up by Mr. Andrew Crosse, at Broomfield, near Taunton. This gentleman collected the electricity by means of insulated wires supported on poles fixed to lofty trees, and carried quite round his grounds. The insulators were so constructed as to be as little as possible affected by the damp, and the length of the exploring wire, as it was called, was upwards of 2,000 feet. These wires terminated in a large insulating conductor placed on his table, near which was another ball, the distance of which could be regulated by means of a screw. By this the fluid escaped to the ground.

The effects produced by this apparatus during stormy weather were said to be awfully grand. A brilliant stream of fire, almost too bright to look at for any length of time, passed between the balls, and the sound was compared by some to the rattle of small fire-arms. His battery consisted of fifty large jars, and required more than 200 turns of a 20-inch plate machine to charge it; yet, when connected with this conductor, it has been charged and discharged as many as twenty times in a minute.

seen.

From observations made with apparatus of this kind, we find that with a clear sky the air is usually charged with positive fluid, the, intensity being greatest at sunrise and sunset. When, however, a large thunder-cloud passes over the apparatus, a series of very strange and interesting phenomena As soon as the edge of the cloud is over the wires, a number of discharges take place; there is then a short pause, and another series of discharges, the electricity being of the opposite kind; then another interval; and in this way there appear to be a number of zones in the cloud, alternately charged with positive and negative fluid. These increase in intensity as they approach the centre of the cloud, and when this part is over the conductor the effect is almost fearful to gaze upon. We must not, however, dwell further upon this here; a fuller account can be found in some works on the subject. One very simple mode of collecting the atmospheric electricity is to fix a metal clip at the end of a fishing-rod. Place

in this some lighted amadou, or a sponge dipped in spirit and lighted, and let a wire connected with it run along the rod. If now it be made to project out of an upper window and carefully insulated, the smoke or flame will collect the electricity, and it may then be examined.

During a fog the air is usually very highly charged, and though all insulators must become damp and therefore imperfect, great quantities of electricity are frequently collected. The whole theory of fogs is as yet very imperfectly understood, but it seems highly probable that electricity is an important agent in their production, and some have even gone so far as to believe that as our acquaintance with their electrical phenomena increases, we may even be able to disperse or prevent them. Though this idea appears highly extravagant, it is still clear that this is an important field for scientific inquiry.

The investigation of the causes of atmospheric electricity has engaged the attention of many scientific men, and many dif ferent hypotheses have been started. One of the principal sources appears to be the evaporation of water from the surface of the earth. The production of electricity in this way is shown by placing a heated platinum capsule on a condensing electroscope, and placing a few drops of water in it. As soon as it has evaporated, and the collecting plate is raised, the leaves will diverge slightly. If, however, distilled water be employed, no effect will be produced.

Clouds are in general charged with electricity, but in storm clouds the charge is much more intense, and if two such clouds oppositely charged approach one another the electricity darts from one to the other in the form of lightning. This kind of lightning is by far the most common, and is not dangerous. At times, however, the clouds are highly charged with similar electricity, and the electric fluid then darts to the earth, often striking any lofty or pointed object that happens to be near. A violent shock is sometimes produced by what is called the return stroke; this is produced by the action of induction. When a highly-charged cloud is near any object, the latter becomes charged by induction with the opposite electricity. As soon, however, as the cloud is discharged this inductive action suddenly ceases, and the result is a shock. A similar effect is felt on standing near the conductor of a very large machine while sparks are being drawn from it. If we watch the motions of the clouds during a thunder-storm, we shall see their attractions and repulsions giving signs of electric action. We have seen that pointed and lofty objects attract electricity, and thus we shall understand the action of lightning conductors. To lofty buildings there is usually affixed a metallic rod, ending in a point, or series of points, projecting some way above the highest part of the building. The lower end of this is connected with a plate of metal buried in the ground, or placed in a reservoir of water. If then a lightning-cloud passes over the building the fluid usually passes off silently by this, and is conveyed to the ground.

It is found in practice that a conductor protects all around it within a radius of about double its own height; for instance, if it projects twelve feet above the roof of a building it will protect all around a space included within a radius of twentyfour feet.

Sometimes a lateral shock is experienced, that is, a portion of the fluid darts off to a good conductor near by. To avoid this, all pieces of metal near the conductor should be placed in metallic connection with it. On account of this kind of discharge it is very dangerous to stand under a tree during a storm, as the fluid may strike the tree, and then leave that to enter the body of the person standing under it. It is always safer then to stand at a little distance from it; disregard of this has cost the lives of many who, in seeking shelter from the rain, have been struck. The neighbourhood of water should also be avoided.

The Aurora Borealis, or Northern Lights, is another very beautiful manifestation of aërial electricity. It is apparently caused by its presence in the upper and highly rarefied regions of the atmosphere. Faint imitations of it have been produced by passing the current through exhausted tubes, and recent experiments have shown that the presence of certain vapours greatly modifies the results, and produces a more striking resemblance to the grand natural phenomena. In polar regions the aurora is seen much more frequently than in our own latitude, and is much more brilliant.

LESSONS IN MENSURATION.-II. FOLLOWING up our subject from the point at which we left it in the former lesson, we subjoin a few examples in the measurement of sides of right-angled triangles, and then pass on to the consideration of triangles which do not contain a right angle. EXAMPLE 1.-A wall is 30 feet high, and it is required to know what length a ladder must be which shall reach to its top, the foot of the ladder not being able to stand nearer the wall than 14 feet.

The 47th Proposition of the First Book of Euclid gives us at once the means of solving the question. The right angle is formed by the wall and the ground; the ladder is therefore the hypothenuse (see Definitions in "Geometry"), and this is equal to the square root of the sum of the squares of the base and perpendicular, or 30+142 = √900+ 196 √ 1096: 30% feet, approximately.

=

EXAMPLE 2.-A ladder is 45 feet long, and when its foot rests upon the edge of the footpath, which is 12 feet wide, its top just reaches the eave of the roof. What height is this eave from the ground?

In this case the hypothenuse and base are known, and the height of the perpendicular is required. This, by the before-mentioned Proposition, is equal to the square root of the difference of the squares of the base and hypothenuse, or √ 452 122 = √2025—1441881 = 43 feet, approximately. EXAMPLE 3.-The side of a square is 9.774 feet; what is the length of the diagonal?

It is necessary here to observe that being a square, the sides are all equal; the length of the diagonal-which is, of course, the same thing as the hypothenuse of either of the triangles formed by the bisection of the square by the diagonal-will, therefore, be the square root of twice the square of one side. This will be in this case nearly 13-823 feet.

EXAMPLE 4.—The side of an isosceles triangle (see Definitions in "Geometry") is 65 feet, and the base 50 feet; what is the

altitude ?

The student must here remember that the sides of the isosceles triangle being equal, the perpendicular bisects the base; hence we have two right-angled triangles formed, in both of which the hypothenuse and base are equal, each to each; the base of each being one-half of that of the isosceles triangle. The rule for right-angled triangles will then apply, and the altitude or perpendicular will be found to be 60 feet.

EXERCISE 1.

1. The base of a right-angled triangle is 4 ft. 6 in. (54 in.), and the hypothenuse 7 ft. 5 in. (89 in.). What is the height of 2. The base being 513, and the perpendicular 684 ft., what is the hypothenuse?

the perpendicular?

3. The hypothenuse is 2 ft. 10 in. and the base 2 ft. 6 in. What is the perpendicular?

4. What is the side of a square whose diagonal is 8 ft. 5 in. ? 5. A ladder 50 feet long, being placed in a street, reached a window 40 feet from the ground, on one side of the street; but when the ladder rested against the house upon the other side of the street, the position of its foot not being altered, it reached a window 48 feet high. What was the breadth of the street? 6. What is the height of an equilateral triangle whose side

is 1?

We subjoin a few examples having reference to the proportion which exists between the homologous sides of similar rightangled triangles, as explained in Lesson I.

EXAMPLE 1.-Two poles stand upright on level ground; the height of one is 10 feet, and its shadow projected upon the ground by the sun is 15 feet. The shadow of the other pole measures 150 feet. What is its height? Ans. 100 feet.

two of the corresponding angles are equal, no matter how great

the disparity of the triangles as to area, the corresponding sides are all respectively proportional. This we have already noticed in our first lesson with respect to right-angled triangles, and the rule holds good in all similar triangles. Let A B C, a b c (Fig. 9) be two similar triangles, having the two angles at A and B equal to the two angles at a and b; then since the three interior angles of every triangle are equal to two right angles (Euc. I. 32), the angle at c must be equal to the angle at c, and the simi

B

Fig. 9.

lar sides are proportional in each triangle; that is to say, AB is to ac as ab is to ac; hence if two sides of one triangle, and a similar side of another and similar triangle be known, the other similar side of the second triangle is found by proportion. We may here observe that this simple and useful rule is equally applicable with respect to the similar lines of all similar figures, whether plane or solid.

In calculating the length of the third side of a triangle, not right-angled-two being known-it is obvious that the rule of the squares (Euc. I. 47) cannot apply, and for this reason: when

Fig. 10.

B

the angle formed by the sides is variable, the sides which contain that angle may remain the same as to length, whilst the hypothenuse may alter. Let A B C (Fig. 10) be a triangle, of which A C is the hypothenuse; then if the angle ABC be not right-angled, it must be either less than a right angle, as

AB C, or greater than one, as A" B C, and the lines A'B, A" B may remain equal as to length, whilst the hypothenuse a'c will be it is obvious that a' c2 and A" c2 cannot equal the same or equal very unequal to the hypothenuse A" c; hence if A C2 = A B2 + BC, quantities; a different mode of treatment must in this case be adopted, which we shall introduce subsequently. In order to find the area of a triangle it is desirable to ascertain the height of the perpendicular, that is, of the line falling vertically upon the base from the opposite angle; and we will here premise that the base of a triangle means its longest side, except in the case of a right-angled triangle, and that the greatest angle of every triangle is always opposite the greater side. (Euc. I. 18.)

Given the three sides of a tri

angle, it is required to find the height of its perpendicular. Let A B C (Fig. 11) be a triangle of which AC is the base, it is required to find the length of the perpenwill be greater than the segment DA. dicular B D. Let BC be greater than

A C 2

B

Then A C B C+BA:: BC-BA: DC-D A, and 2 + the length of the greater segment D C, which being subtracted from A c gives the lesser segment DA.

Then

We have thus ascertained the position of the point D. in either of the right-angled triangles A D B, C D B we have the two sides A D, A B, and C D, C B, from which, by Euc. I. 47, we find the height of D B.

Next, having given the length of the base, and the height of the perpendicular of a triangle, to find its area. The rule is of the very simplest kind :Multiply the base by half the perpendicular, and the result is the area of the triangle. The reason of this we will prove :-Let A B C (Fig. 12) be a right-angled triangle, right-angled at B B. Complete the parallelogram ABCD; then AC bisects it (Euc. I. 34). Now the area of a square or of a right-angled parallelogram is the two adjacent sides. Hence A BX B C is the area of the parallelogram A B C D, but this is double the area of the triangle A B C (Euc. I. 41). Hence if ABX BC= area of parallelogram ABCD, AB X = area of triangle A B C.