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332 upon him, and on the night of May 18, 1822, he was made Emperor. In his address to the people he said, "If, Mexicans, I do not secure the happiness of the country; if at any time I forget my duties, let my sovereignty cease." He was crowned by the bishop, but with his own hands he placed the diadem on the brow of Doña Ana. An imperial household was established with imposing splendor, and money was coined in his image. He also instituted the Order of Guadalupe, a return to the days of chivalry, and designed to add to the prestige of the government. But "uneasy lies the head that wears a crown," and Iturbide was no exception to the truth of the apothegm. Only nine months from his coronation, pressure of circumstances and political changes forced him to abdicate. A sentence of exile was pronounced against him, and three months later, with his family, he was on his voyage to Italy.

To the soldier accustomed to a life of action, exile was intolerable; and possessed of an irresistible desire to return, within a year he made the homeward bound journey which proved fatal. A new and hostile government was in power, and Iturbide had lost his old influence. Not knowing the stern attitude of the government toward him, he landed July 14, 1824, at Soto la Marina, on the gulf coast; and scarcely had he touched his mother soil when he found himself a prisoner.

General Garza, the military commander, unwilling to act on his own responsibility, referred the matter to the State Congress of Tamaulipas, then in session at Padilla. With much show of respect and seeming confidence the ex-Emperor was conducted thither. He arrived late at night, hopeful and unsuspicious, having himself been placed by Garza in command of the escort which accompanied him. The next morning he was informed that he must prepare for death that afternoon. He remonstrated, asserting his innocence of any desire to disturb the existing order of things, and referring in proof of this to the presence of his family on shipboard. On finding the decree inexorable, he said, "Tell General Garza I am ready to die, and only request three days to prepare to leave this world as a Christian."


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334 time for Morelia to remember her two most distinguished sons.

The title of Liberator was conferred upon Iturbide in 1853, nearly thirty years after his death, and two years later the anniversary of his death was declared a public holiday. On that day a grand mass is celebrated in the cathedral of Mexico for the repose of his soul.

The ex-Emperor left a wife and eight children, but only the two youngest and Doña Ana accompanied him on his fateful return voyage, the others being left at school in England. The widow went first to New Orleans, afterward lived in Washington, then in Baltimore, finally taking up her permanent residence at Philadelphia, where in 1861 the once beautiful Dona Ana ended her eventful life, and now rests with several of her children in a vault of St. Mary's Church in that city.

The Princess Josefa, the only surviving child of the Emperor, resides in the City of Mexico. She remembers the coronation of her father and the pomp of court life which followed during his short reign. It was my pleasure to make her acquaintance, and I found her a woman of rare conversational gifts as well as great personal charm of manner. She is remarkably well preserved, and still shows a vigorous and cultivated intellect; is a fine linguist, and possesses a vast amount of historical information. But the one who connects the past with the present is Prince Angel de Iturbide. He attended the Jesuit College at Georgetown, D. C, where as a school-boy he met and loved Alice Green, the lovely daughter of Nathaniel Green, of that city. The wooing was persistent, and finally this charming and accomplished woman became his wife. In the course of time the laws which had banished Doña Ana and her family relented, and the Iturbides were allowed to return to Mexico.

Now comes an old, old story, but one which loses nothing by familiarity. In the checkered fortunes of Mexico, a prince of the house of Habsburg and an Austrian archduke was invited by the

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336 Augustin to Havana, and at the same time communicated with Madame Iturbide, who joyfully met and received again to her tender heart her idolized boy. He is now a strikingly handsome young man, twenty-three years of age, six feet in height, and possessing wonderful physical strength. He has a finished education, both European and American, and is an accomplished linguist. He is also a lover of scientific knowledge, and exceptionally well read in history. Added to these natural and acquired advantages, he has artistic tastes, sketches from nature, and is skilled in music. In 1885 he was awarded the gold medal at the college at Georgetown, D. C, for the best oration delivered at the closing exercises. The hero of a romantic story, he appears unconscious of the notice he has attracted, and retains his modest demeanor and genial disposition, with the dignity and social graces which render his society delightful to all who come in contact with him. On his handsome country estate he leads a business life, and never seems happier than when there, dressed in his buckskin suit and silver-decked sombrero, and mingling freely among his employees, who adore him. The minutest detail of hacienda life claims his careful attention, showing a happy adaptability to circumstances.

The elegant residence of the Iturbides at the capital stands on the grand Pasco, immediately to the right of the statue of Carlos IV. Both there and at their hacienda of San Miguel Sesma, I have enjoyed their graceful hospitality and unrestricted friendship. On these occasions Madame Iturbide related many interesting incidents and reminiscences of her boy's early life. Among them, to me, one of the most amusing was the manner in which Augustin, when a little more than four years old, spoke his first English. His cousin, Plater Green, a few months older, fell from a tree, when Augustin ran to his parents, crying out: "Plater he up de tree—Plater he down de tree—Plater he no cry—Plater he one very man!" After this he would speak no more Spanish. Although brought up according to the Mexican custom of dependence on a servant, he early manifested the desire to throw off such bondage and prove his self-reliance. At the


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340 of the statesmen and patriots who bore a conspicuous part in those thrilling scenes. All who are linked by lineage or ties of consanguinity to the heroes of the revolution, preserve sacredly every reminder and relic of their progenitors. Amid such surroundings, my desire for information was stimulated, and the impressions then received remain among the choicest treasures of memory garnered during my sojourn in old Mexico.

Vicente Guerrero was one of the leading spirits of the revolutionary period, and is revered in the history of his country as a man of unyielding patriotism, strict integrity, and stanch loyalty to its cause. After the death of Morelos, the germs of independence were kept alive and nurtured by Guerrero, who operated in the southwest, and was the most conspicuous figure among the insurgents when joined by Iturbide.

In the conflicts which have been waged on Mexican soil, guerrilla warfare has always borne a leading part, the inaccessible mountain fastnesses yielding immunity from danger of pursuit. This was the method pursued by the leaders after the fall of Hidalgo, Morelos, and Matamoras. When at last independence was achieved, Guerrero took an active part in every important movement until his death

He was the third president of the republic, and had served only a short time when he was deposed by Bustamente, then vice-president. He retired to his country estate, Tierre Colorado, in the vicinity of Tixtla; but being informed of a plot against his life, he left there and joined Alvarez, then in revolt against the government which had succeeded that of Guerrero. Fearing his influence, his death was determined on, and when, despite the warnings of Alvarez, he went to


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342 politics. Picaluga, the Genoese, was sentenced by his government to death, and mulcted in heavy damages; but as he could not be found, he escaped punishment. Gonzales, who received the hapless Guerrero at Guatulco, died miserably, a slow, torturous death.

Many tributes to the public and private virtues of Guerrero may be found in various places; and his name is perpetuated in that of one of the States of the Republic. It was said of him that "his modesty overshadowed his intelligence to the extent of not allowing him to enjoy the fruits of his services as his talents deserved."

Guerrero left a wife and one child, a daughter, who became the wife of Mariano Riva Palacio, afterward one of the most distinguished lawyers and public men of his time. Their son is General Vicente Riva Palacio, so often mentioned in these chapters.

I would like to dwell at length on the Bravos—Leonardo, the father, and Nicolas, the son. They loved their country with exalted patriotism, and devoted their lives to its liberation. Nicolas is spoken of by historians as one of the noblest specimens of manhood that the times produced. They were no less attached to each other than to their country.

After the battle of Cuantla, the father was taken prisoner, tried, and condemned to be shot. Venegas, the viceroy, so highly appreciated his abilities that he offered Bravo his life if he would induce his brothers and Nicolas to join the royalists. But liberty was his watchword; he scorned the offer, and paid the forfeit. A number of Spanish prisoners had been offered in exchange for him, but the viceroy, appreciating the value of a Bravo, had declined in his turn.

The grief of Nicolas for his father was deep and lasting; but even under this great sorrow his magnanimity shines forth grandly. He had then in his camp, as prisoners, three hundred Spaniards, many of them wealthy and influential men. His power over them was absolute; and had he taken their lives in retaliation for his beloved father's death, perhaps justice and the usages of war would have said,

"Well done!" But hear his noble words to them:

"Your lives are forfeit. Your master, Spain's minion, has murdered


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The New International Encyclopædia/Hemlock-tree

HEMLOCK-TREE, or. A name applied to the coniferous trees which belong to the genus Tsuga, especially to Tsuga Canadensis. This is a large, graceful tree with much the habit and appearance of some spruces. It is found from Nova Scotia to Alabama and Georgia, and west to Wisconsin and Minnesota. The leaves are one-half inch long, bright green above and silvery beneath. The cones, which have few thin bracts, are scarcely longer than the leaves. The wood, light, soft, brittle, and light-red in color, is extensively used in building. It warps badly when exposed. The bark is largely employed in tanning leather, for which purpose it is preferred and more extensively used than any other American product. Hemlock oil is distilled from the branches and leaves, and pitch is obtained from the trees. There are numerous cultivated varieties which are very ornamental, especially while the trees are young. Tsuga Caroliniana is a second species, which occurs locally in Virginia, North Carolina, and Georgia. Other species are found in Asia. See.


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The New International Encyclopædia/Hemlock

HEMLOCK (AS. hemlic, hymblicæ, hemlock), Conium. A genus of plants of the natural order Umbelliferæ, the members of which have compound umbels of small white flowers, small involucres and involucels, the former consisting of several small leaflets, the latter of three leaflets all on one side; the limb of the calyx merely rudimentary, and a compressed ovate fruit with five prominent wavy ridges and no vittæ. The best known and only important species is the common or poison hemlock (Conium maculatum), which grows by waysides on heaps of rubbish, and in other similar situations throughout Europe, in some parts of Asia, and naturalized in North America and Chile. It has a root somewhat resembling a small parsnip; a round, branched, hollow, bright-green stem, two to seven feet high, generally spotted with dark purple; large tripinnate, dark shining green leaves with lanceolate pinnatifid leaflets. All parts of the plant are destitute of hairs. The whole plant has a nauseous smell, particularly if rubbed or bruised. The leaves and fruits are employed in medicine. The leaves should be gathered just before the time or at the commencement of flowering, and after the removal of the larger stalks they should be quickly dried by a heat not exceeding 120°. As, however, they sometimes yield little or none of the active principles in the plant, the fresh leaves are preferred. The whole plant contains the active principles, and many fatal cases of poisoning have been attributed to eating the roots under the mistaken idea that they were parsnips.

The uses of hemlock in medicine are few and unimportant and depend chiefly upon its action upon the motor nerves, beginning with their end-organs. Although large doses cause complete paralysis by their action upon the peripheral nerves, sensation and consciousness are not affected; death finally occurs by extension of the paralysis to the muscles of respiration. In large or poisonous doses it sometimes gives rise to coma, and sometimes to convulsions or violent delirium. Among the ancient Greeks poisoning by hemlock was a common mode of death for condemned criminals, but whether it was the juice of the common hemlock or the water-hemlock that was used is unknown. The strength of its preparations is very variable, as its active principles are volatile. Practically the only use of the drug is in tetanus, hydrophobia, strychnine poisoning, and other convulsive disorders. Even in these it is rarely used, and

only to prevent exhaustion by lessening muscular contractions. Water-hemlock, or cowbane (Cicuta virosa), is also an umbelliferous plant, of a genus having vaulted umbels, a live-toothed calyx, and almost globose fruit, each carpel with five broad flattened ribs and evident single vittæ. Water-hemlock grows in ditches, the margins of ponds, and wet grounds in Europe and the north of Asia. It has a large fleshy while root, covered externally with fibres; an erect much-branched stem, two to five feet high; tripinnate leaves, with linear-lanceolate regularly and sharply serrated leaflets, one general involucre or only a single small leaflet, partial involucres of many short narrow leaflets, and white flowers. It is a virulent narcotic acrid poison. Another species, Cicuta maculata, is common in North America, growing in many places. It has a spotted stem, like that of true hemlock, the name of which it very generally receives in North America. The leaves are triternate, the leaflets ternate. It is a very poisonous plant, and is the cause of many deaths. Cicuta, in Latin, seems to have been the name of the same plant called concion by the Greeks, but it is not known whether this or the previous plant was so denominated. See.


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The New International Encyclopædia/Hemp, Bowstring

HEMP,. The fibre produced by Sansevieria spp., a genus of Liliaceæ with representatives in tropical regions of both hemispheres. The employment of the fibre for making bowstrings led to this name. The plants are stemless perennials which grow wild in the jungles and increase by runners. The leaves, which yield readily to treatment in the fresh state, are thick and fleshy, sword or lance-shaped, and rich in fine lustrous fibre which ranges from two and one-half to seven feet in length, does not rot readily in water, and by experiment has been found to be the strongest and best-fitted fibre for deep-sea sounding. The plant, which is propagated readily from cuttings, requires a good rich soil in which, under favorable conditions, it will reach its full growth in one year. Ordinarily, however, it does not acquire its full size until the second season, and some species do not yield a crop until the third year. When once the land is stocked with it a full crop may be expected from the roots within twelve months after cutting. It does not appear to exhaust the soil materially, and will continue to make vigorous growth for a number of years in the same place. The yield is about 50 pounds of fibre per ton of green leaves. The total yield per acre has been reported by experiments to be nearly two and a-half tons of fibre annually.


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The New International Encyclopædia/Hemp, Indian

HEMP,. See.


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 ARMENIA

I

Lord Christ, when yet thy face was young

In heaven, and thy witnesses were few,

Humble thy Kingdom here, nor yet grace drew

Emperors to the breast where Lazarus clung,—

When round a dying world thy arms were flung,—

Armenia first unto thy mercies flew,

To the pure gospel through all ages true,

And Him, whose sorrows on the world's cross hung.

She, who beheld the glorious covenant,

When o'er the Flood, at the Creative Word,

Bright above Ararat sprang the bow in heaven,—

What to her agony will thy pity grant?

For unto her through faith in thee, O Lord,

The thorny crown of Christendom is given.

220


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The New International Encyclopædia/Hemp, Manila

HEMP,, or. The fibre of a species of plantain or banana (Musa textilis), a native of the Philippine Islands, where it is extensively cultivated. It is most widely grown in the southern part of Luzon, and less extensively on the islands of Samar, Leyte, Cebu, and Mindanao. The fibre is white, easily separated, and very tenacious. The harder and stronger outer fibre is used for cordage, the finer inner fibres for weaving delicate fabrics, and the intermediate layers for manufacture of web-cloths and gauzes. From the finer sorts of the fibre, veils, crapes, robes, and women's hats of great beauty are made; also various articles of men's wear, as shirts and suitings of remarkable durability. Manila hemp is largely used in the United States for making binding twine and cordage. Manila cordage is now used for every purpose for which rope is employed. The best conditions for the growth of abaca are shade and abundant moisture, with good drainage. The finest growth is obtained on the slopes of the volcanic mountains. A new plantation is generally made of young shoots, which very quickly throw out suckers. Occasionally the plants are raised from seed. The cultivation of the plant is simple. Weeds and undergrowth must be kept down during the first season, but after that the plants will exterminate other growths. Plants raised from year-old seedlings need at least two years, while those raised from suckers require four years before they will produce fibre of value.

The abaca is cut just before flowering, near the roots, split open longitudinally, and the central peduncle separated from the sheathing layers of fibre. The fibrous coats are dried two or three days in the shade, and are then scraped until only the fibres remain. Two men will cut and scrape about twenty-five pounds of the fibre in a day. It takes about 3200 plants to produce a

ton of 2240 pounds of fibre. The yield is about one and one-third tons of fibre per acre annually. The output of abaca is fairly uniform under natural conditions. The exports for several years preceding the cession of the islands by Spain averaged about 100,000 tons annually.


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Rh II

, all ye nations, myrrh and frankincense,

As when, with gold and many an orient gem,

About the cradled child of Bethlehem

Like heaven the holy stable glittered, whence

Issued salvation! Pour the providence

Of earthly kingdoms at the feet of them

Who would a world-wide flood of sorrow stem

And, Christ-like, feed the multitude immense!

Nor think Armenia only bears the Cross

Through deserts wild and up her mountain-chain;

But every nation climbs its Calvary,

And hath its consecration; earthly loss

Thousands on thousands find is heavenly gain:

So the world-soul renews humanity.


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 AN EASTER ODE

1918

Inscribed in Memory of Lieutenant Edward Bedinger Mitchell

DULCE ET DECORUM EST PRO PATRIA MORI

I

Spring, thy rosy tides

O'er earth's pale shoulder glow;

From heavenly peaks, down Europe's sides,

The torrent sunbeams flow;

Across the verdure-belted zones what ceaseless seasons go!

All, all, indifferent to human woe!

The sea with corpses blossoms, as of old

On the bright Salaminian bay

Ere the gray waste, unrolled,

On the wide-wanderer's eyes flung dim Pacific spray;

Immeasurable spreads afar

The battle-tossing plain of war,

And of fair cities makes a gaunt volcanic scar;

From up-torn realms untenanted

The beasts and birds affrighted fled; 222


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Rh Prone, where the sire his life-blood shed,

The mother on the child lies dead;

The torch, the axe, the bomb, the shell

Paint earth and heaven in hues of hell;

Famine, massacre, slavery fall

On man in horrid carnival.

Great is thy triumph, modern age!

Progress thy bane, science thy scourge,

In sea and air new wars to wage,

And aye to evil fates, incessant, urge

Man's miserable race, on ruin's awful verge!

II

Meanwhile, on blue-horizoned shores, against Floridian skies,

Lone, white cranes, standing, fish; from sunset-colored caves

The darting mullet hues the shadow-haunted waves;

In pale, pellucid depths the rude crustacean lies.

There, with the dædal earth

The great Creator toys;

A thousand shapes of mirth,

A million vivid joys,

Like grains of coral sand,

Drop from His listless hand.—

How should man understand?


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 the gulf, it possesses no deep water port. North of Katif it is desert and only inhabited by nomads; at Katif, however, and throughout the district to the south bordering on the Gulf of Bahrein there are ample supplies of underground water, welling up in abundant springs often at a high temperature, and bringing fertility to an extensive district of which El Hofuf, a town of 15,000 to 20,000 inhabitants, is the most important centre.

South-western Arabia, from the twenty-first parallel down to the Gulf of Aden, including the Taif district of Hejaz, Asir and Yemen, forms one province geographically. Throughout its length it consists of three zones, a narrow coastal strip, rarely exceeding 20 m. in width, a central mountainous tract, embracing the great chain which runs parallel to the coast from near Taif to within 50 m. of Aden, and an inner plateau falling gradually to the north-east till it merges in the Nejd steppes or the sands of the great desert.

The lowland strip or Tehama consists partly of a gravelly plain, the Khabt, covered sparsely with acacia and other desert shrubs and trees, and furnishing pasturage for large flocks of goats and camels; and partly of sterile wastes of sand like the Ramla, which extends on either side of Aden almost from the seashore to the foot of the hills. The Tehama is, however, by no means all desert, the mountain torrents where they debouch into the plain have formed considerable tracts of alluvial soil of the highest degree of fertility producing in that warm equable climate two and even three crops in the year. The flood-water is controlled by a system of dams and channels constructed so as to utilize every drop, and the extent of cultivation is limited more by the supply of water available than by the amount of suitable soil. These districts support a large settled population and several considerable towns, of which Bet el Fakih and Zubed in the western and Lahej in the southern Tehama, with 4000 to 6000 inhabitants, are the most important. There are signs that this coastal strip was until a geologically recent period below sea-level; and that the coast-line is still receding is evidenced by the history of the town of Muza, once a flourishing port, now 20 m. inland; while Bet el Fakih and Zubed, once important centres of the coffee trade, have lost their position through the silting up of the ports which formerly served them.

The jebel or mountain-land is, however, the typical Yemen, the Arabia Felix of the ancients. Deep valleys winding through the barren foothills lead gradually up to the higher mountains, and as the track ascends the scenery and vegetation change their character; the trees which line the banks of the wadi are overgrown with creepers, and the running stream is dammed at frequent intervals, and led off in artificial channels to irrigate the fields on either side; the steeper parts of the road are paved with large stones, substantially built villages, with their masonry towers or dars, crowning every height, replace the collection of mud walls and brushwood huts of the low country; while tier above tier, terraced fields cover the hill slopes and attest the industry of the inhabitants and the fertility of their mountains. On the main route from Hodeda to Sana the first coffee plantations are reached at Usil, at an altitude of 4300 ft., and throughout the western slopes of the range up to an altitude of 7000 ft. it is the most important crop. Jebel Haraz, of which Manakha, a small town of 3000 inhabitants is the chief place, is described by Glaser as one vast coffee garden. Here the traveller ascending from the coast sees the first example of the jebel or highland towns, with their high three-storeyed houses, built of quarried stone, their narrow façades pierced with small windows with whitewashed borders and ornamented with varied arabesque patterns; each dar has the appearance of a small castle complete in itself, and the general effect is rather that of a cluster of separate forts than of a town occupied by a united community.

The scenery in this mountain region is of the most varied description; bare precipitous hill-sides seamed with dry, rocky watercourses give place with almost startling rapidity to fertile slopes, terraced literally for thousands of feet. General Haig in describing them says: “One can hardly realize the enormous labour, toil and perseverance that these represent; the terrace walls are usually 5 to 8 ft. in height, but towards the top of the mountains they are sometimes as much as 15 or 18 ft.; they are built entirely of rough stone without mortar, and I reckon that on an average each wall retains not more than twice its own height in breadth, and I do not think I saw a single break in them unrepaired.”

The highest summits as determined by actual survey are between 10,000 and 11,000 ft. above sea-level. J. Sabur, a conspicuous mass in the extreme south, is 9900 ft., with a fall to the Taiz valley of 5000 ft.; farther north several points in the mountains above Ibb and Yarim attain a height of 10,500 ft., and J. Hadur, near the Sana-Hodeda road, exceeds 10,000 ft. From the crest of the range there is a short drop of 2000 or 3000 ft. to the broad open valleys which form the principal feature of the inner plateau. The town of Yarim lies near its southern extremity at an altitude of about 8000 ft.; within a short distance are the sources of the W. Yakla, W. Bana and W. Zubed, running respectively east and south and west. The first named is a dry watercourse ultimately joining the basin of the W. Hadramut; the two others run for a long distance through fertile valleys and, like many of the wadis on the seaward side of the range, have perennial streams down to within a few miles of the sea. Sana, the capital of Yemen, lies in a broad valley 7300 ft. above sea-level, sloping northwards to the W. Kharid which, with the Ghail Hirran, the sources of which are on the eastern slopes of J. Hadur, run north-eastward to the Jauf depression. The Arhab district, through which these two great wadis run, was formerly the centre of the Himyar kingdom; cultivation is now only to be found in the lower parts on the borders of the watercourses, all above being naked rock from which every particle of soil has been denuded. In the higher parts there are fine plains where Glaser found numerous Himyaritic remains, and which he considers were undoubtedly cultivated formerly, but they have long fallen out of cultivation owing to denudation and desiccation—the impoverishment of the country from these causes is increasing. Eastward the plateau becomes still more sterile, and its elevation probably falls more rapidly till it reaches the level of the Jauf and Nejran valleys on the borders of the desert. The water-parting between central and southern Arabia seems to be somewhere to the south of Nejran, which, according to Halévy, drains northward to the W. Dawasir, while the Jauf is either an isolated depression, or perhaps forms part of the Hadramut basin.

Farther north, in Asir, the plateau is more mountainous and contains many fertile valleys. Of these may be mentioned Khamis Mishet and the Wadi Shahran rising among the high summits of the maritime chain, and the principal affluents of the Wadi Besha; the latter is a broad well-watered valley, with numerous scattered hamlets, four days’ journey (perhaps 80 m.) from the crest of the range. Still farther north is the Wadi Taraba and its branches running down from the highland district of Zahran. The lower valleys produce dates in abundance, and at higher elevations wheat, barley, millets and excellent fruit are grown, while juniper forests are said to cover the mountain slopes. In Yemen this tree was probably more common formerly; the place-name Arar, signifying juniper, is still often found where the tree no longer exists.

The western coast of Yemen, like that of Hejaz, is studded with shoals and islands, of which Perim in the Straits of Bab-el-Mandeb, Kamaran, the Turkish quarantine post, 40 m. north of Hodeda, and the Farsan group, off the Abu Arish coast, are the principal. Hodeda is the only port of any importance since the days of steamships began; the other ports, Mokha, Lohaia and Kanfuda merely share in the coasting trade. The south coast is free from the shoals that imperil the navigation of the Red Sea, and in Aden it possesses the only safe natural harbour on the route between Suez and India. Several isolated volcanic hills crop out on the shore line between Aden and the straits; the most remarkable are J. Kharaz, 2500 ft., and J. Shamshan, 1700 ft., at the base of which Aden itself is built. In both of these the crater form is very clearly marked. A low maritime plain, similar to the Tehama of the western coast, extends for some 200 m. east of the Straits of Bab-el-Mandeb, backed by mountains rising to 7000 ft. or more; farther east the elevation of the highland decreases steadily, and in the Hadramut, north of Mukalla, does not much exceed 4000 ft. The mountain chain, too, is less distinctly marked, and becomes little more than the seaward escarpment of the plateau which intervenes between the coast and the Hadramut valley. This valley runs nearly east and west for a distance of 500 m. from the eastern slopes of the Yemen highlands to its mouth on the Mahra coast near Sihut. The greater part of it is desert, but a short stretch lying between the 48th and 50th meridians is well watered and exceptionally fertile. This begins a little to the east of Shabwa, the ancient capital, now half buried in the advancing sand, and for a distance of over 70 m. a succession of villages and towns surrounded by fields and date groves extends along the main valley and into the tributaries which join it from the south. Shibam, Saiyun and Tarim are towns of 6000 or more inhabitants, and Hajren and Haura in the W. Duwan are among the larger villages. Himyaritic remains have been found here and in the W. Mefat which enters the Gulf of Aden near Balhaf. A few small fishing villages or ports are scattered along the coast, but except Mukalla and Shihr none is of any importance.

The Gara coast was visited by the Bents, who went inland from Dhafar, one of the centres of the old frankincense trade, to the crest of the plateau. The narrow coastal strip seems to be moderately fertile, and the hills which in places come down to the seashore are covered with trees, among which the frankincense and other gum-bearing trees are found. On the plateau, which has an altitude of 4000 ft., there is good pasturage; inland the country slopes gently to a broad valley beyond which the view was bounded by the level horizon of the desert.

(q.v.) includes all the south-eastern corner of the peninsula. Its chief feature is the lofty range of J. Akhdar, 10,000 ft. above sea-level. Like the great range of western Arabia, it runs parallel to the coast; it differs, however, from the western range in that its fall on the landward side is as abrupt and nearly as great as on its seaward side. Its northern extremity, Ras Musandan, rises precipitously from the straits of Hormuz; farther south the range curves inland somewhat, leaving a narrow but fertile strip, known as the Batina coast, between it and the sea, and containing several populous towns and villages of which Sohar, Barka and Sib are the chief. Muscat, the capital of the province and the principal port on the coast, is surrounded on three sides by bare, rocky hills, and has the reputation of being the hottest place in

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224 III

O Easter moon that glorious

In highest heaven dost roll,

What saw you on the Caucasus

Great with Prometheus' soul?

Where Calvary's shining road makes up from the dark vale below,

Saw you thorn-crowned beneath a Cross a man of sorrows go,

The Sufferer, who never dies, but bears the whole world's woe?

Saw you from Athens' ghostly hand the torch of truth burn bright,

That spreads within the mind the world where shall be no more night?

Saw you the Tiber, Seine and Thames, the floods that shake the North,

Pour inexhaustibly their hosts of stern-faced freemen forth?

Far as your circling light below hath on our oceans broke,

Saw you the little acorns grown, blown from the English oak,

The tree of liberty, that laughs amid the thunder-stroke?


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The New International Encyclopædia/Hemp, Sisal

HEMP,. The fibre of Agave rigida sisalana, order Ainaryllideæ, native of Yucatan, Mexico, Central America, and West Indies, and closely related to the century plant. It has been introduced into the Bahamas and Florida. As a cordage fibre it is second only to Manila hemp in strength. Its straight smooth strands of fibre become white upon drying without washing. The plants yield best on a gravelly, rocky soil in comparatively arid districts only a few feet above the level of the sea. It thrives upon the Florida Keys, upon the almost naked coral rock, but will not thrive when even slightly shaded. Plants set out at 18 inches high from the nursery will produce leaves fit for cutting in three years. The life of a plant when undisturbed is six or seven years, after which it sends up its blossom-stalk and then dies. Cutting, however, extends its life fifteen to twenty years. No special cultivation is needed except that the land be kept clean and the suckers kept down. The cutting of the leaves is done in Yucatan by Indians, who use a heavy-bladed, sabre-like knife called a machete. The spine at the leaf-end is cut off and the leaves bundled for removal to the machines. On large plantations they are transported by steam-power over tramways which reach different parts of the estate. The fibre is separated from the fresh leaf by a specially constructed machine called a raspodor. The annual yield of fibre is from 1000 to 1470 pounds per acre, 50 to 70 pounds of fibre being derived from 1000 leaves.


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Rh And Paris, Honor's fount—O name that never time forgets!—

Look you! how high in our sad heavens her ray of glory jets!

Look! as your crescent horn but late filled its dark curve with light,

So grows America on earth amid the nations bright!—

Or is it, crystal sphere serene that hast no mortal stain,

You do not mind, at all, these things, which man has done in vain?

Oh, can it be, then, nature's law

That her the vision fails,—

The dream divine, and holy awe

That in man most avails?

And know you not, celestial orb

The light men's souls from you absorb

Beholding, when dark deaths they risk,

With highest instincts in accord,

How pure in heaven your golden disk

the Risen Lord?

IV

Upon the border of eternity—

As some Greek runner, on high mountain ways,

Whom now at eve his speed of morn delays,

Hears the far rote of his own native sea—


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The New International Encyclopædia/Hemp, Sunn

HEMP,. The fibre derived from the bark of Crotalaria juncea, order Leguminosæ, a native of India. The plant has been in cultivation from time immemorial upon the high, sandy lands less suited for the more exacting crops. Seed is generally sown in April or May, and in August the plant, if grown for its fibre, is pulled or cut close to the ground, laid in long rows till the leaves begin to rot and separate from the stalks, which are then steeped in water for a few days, till the bark separates freely. The average yield is about 640 pounds of fibre per acre. The fibre is not so strong as hemp, but good cables, canvas, and cloth are made of it. It is now exported in considerable quantity, and is known by various names, as brown hemp, Bengal hemp, etc. Jubbulpore hemp (Crotolaria tenuifolia) is considered by some authorities to be a variety of Crotalaria juncea. See.


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226 I harken unto deathless voices rolled

From the great deep, and silent lyres of old;

And with the sound thereof my lips grow bold.

Man's is another world

Wherein the spirit flies;

Truth at his heart impearled,

A thousand deaths he dies.

O wake again, Tyrtæan lyre

That flung the world's first tyrants low!

Heap up thy urn with holy fire

That now doth in all peoples glow!

Once more the dreadful trumpet sound

Of freedom, Macedonian mound!

Thou, gray Thermopylæ, arise!

Who lifted first on human eyes

Victorious shields of sacrifice,—

And old Simonides thy glory crowned,

Leading the poets' bright, immortal choir.

Still rolls aloft the heroic hymn

Of men, when light and life grow dim.

O sacred bands, dear to the lyre's blest breath

That, ever resonant with noble death,

Sweeps eagle-borne round glory's cloudy wreath,

A thousand dawns we sang you to the fight,

A thousand victories sang you home at night!


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 {|style="float: left;" front of the kisi. The expectancy now is intense. All eyes are fixed in the direction from which the priests will appear. The sun sinks lower and the evening colors steal into the landscape, but no one notices them. "Here they come!" The grand entry of the Antelope priests causes a sensation. With bare feet, and their semi-nude bodies streaked with white paint; a band of white on the chin from mouth to ear, rattles of tortoise shell tied to the knee, embroidered kilts of white cotton fastened around the loins, necklaces of shell and turquoise, and fox skins hanging behind from the belt, these priests present a startling though not unattractive appearance. At the head of the file comes the Antelope Chief bearing his tiponi or sacred badge across his left arm. Next comes the bearer of the medicine bowl. All the other priests carry a small rattle in either hand. With stately mien, and looking to neither right nor left, the Antelope priests pass four times around the plaza to the left, each sprinkling sacred meal and stamping violently upon the plank in the ground in front of the kisi. The hole in the middle of the plank is the opening into the under-world and the dancers stamp upon it to inform the spirits of their ancestors that a ceremony is in progress. Fortunate is the man who breaks the board with his foot! When the circuit is made, the Antelope priests line up in front of the kisi facing outward; thereis a hush and the Snake priests enter.
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Rh Look up, ye hosts! o'er heroes when they die

Opens in heaven another climbing sky!

Sweet is your memory here, and fresh with tears

That wash from shining eyes our mortal fears.—

Peace at the last, and moods all joys above,

Calm thoughts that from just reason take their birth!

Truth at the last, and liberty, and love

Shall, like your glory, fill the ensanguined earth!


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The New International Encyclopædia/Hemp-agrimony

HEMP-AGRIMONY. A European medicinal plant. See.


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 The grand entry of the Snake priests is dramatic to the last degree. With majestic strides they hasten into the plaza, every attitude full of energy and fierce determined purpose. The costume of the priests of the sister society of Antelopes is gay in comparison with that of the Snake priests. Their bodies rubbed with red paint, their chins blackened and outlined with a white stripe, their dark red kilts and moccasins, their barbaric ornaments, give the Snake priests a most somber and diabolical appearance. Around the plaza, by a wider circuit than the Antelopes, they go striking the sipapu plank with the foot and fiercely leaping upon it with wild gestures. Four times the circuit is made; then a line is formed facing the line of the Antelopes, who cease shaking their rattles which simulate the warning note of the rattlesnake. A moment's pause and the rattles begin again and a deep humming chant accompanies them. The priests sway from side

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The New International Encyclopædia/Hasenauer, Karl, Baron von

HASENAUER, hä'ze-nou'ẽr,, Baron von (1833-94). An Austrian architect, born in Vienna, where he studied at the academy under Van der Nüll and Siccardsburg, and obtained the first prize in 1854. After having traveled extensively in Germany, Italy, France, England, and Scotland, he entered upon an active career in Vienna, was again awarded the first prize for architecture at the art exhibition of 1864, became a member of the Academy of Fine Arts in 1866, and honorary member of the Institute of British Architects in 1868. Among a number of

private palaces built by him, the Palais Lützow is to be noticed as one of the finest houses in Vienna, but his most important work was the erection of the magnificent Imperial Museums of Art and Natural History (1872-91) in purest Renaissance style, after his own and Semper's designs. He also built the new Hofburg theatre (completed 18S8), and designed the addition to the Imperial palace. He was architect for the buildings of the Vienna Exposition in 1873, and was raised to a baronetcy in the same year.


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 to side, sweeping their eagle-feather snake whips toward the ground; the song grows louder and the lines sway backward and forward toward each other like two long, undulating serpents. The bearer of the medicine walks back and forth between the lines and sprinkles the charm liquid to the compass points.

All at once the Snake line breaks up into groups of three, composed of the "carrier" and two attendants. The song becomes more animated and the groups dance, or rather hop, around in a circle in front of the kisi, one attendant (the "hugger") placing his arm over the shoulder of the "carrier" and the other (the "gatherer") walking behind. In all this stir and excitement it has been rather difficult to see why the "carrier" dropped on his knees in front of the kisi; a moment later he is seen to rise with a squirming snake, which he places midway in his mouth, and the trio dance around the circle, followed by other trios bearing hideous snakes. The "hugger" waves his feather wand before the snake to attract its attention, but the reptile inquiringly thrusts its head against the "carrier's" breast and cheeks and twists its body into knots and coils. On come the demoniacal groups, to music now deep and resonant and now rising to a frenzied pitch, accompanied by the unceasing sibilant rattles of the Antelope chorus. Four times around and the "carrier" opens his mouth and drops the snake to the ground

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 and the "gatherer" dextrously picks it up, adding in the same manner from time to time other snakes, till he may have quite a bundle composed of rattlesnakes, bull snakes and arrow snakes. The bull snakes are large and showy, and impressive out of proportion to their harmfulness. When all the snakes have been duly danced around the ring, and the nerve tension is at its highest pitch, there is a pause; the old priest advances to an open place and sprinkles sacred meal on the ground, outlining a ring with the six compass points, while the Snake priests gather around. At a given signal the snakes are thrown on the meal drawing and a wild scramble for them ensues, amid a rain of spittle from the spectators on the walls above. Only an instant and the priests start up, each with one or more snakes; away they dart for the trail to carry the rain-bringing messengers to their native hiding places. They dash down the mesa and reappear far out on the trails below, running like the wind with their grewsome burdens. The Antelope priests next march gravely around the plaza four times, thumping the sunken plank, and file out to their kiva. The ceremony is done.

Stay! there is another scene in this drama which may seem a fitting

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 termination. Whoever wishes may go to look on, but not everyone goes. The Snake priests return, go to the kiva and remove all their trappings, come out to the edge of the cliff where the medicine women have brought great bowls of a dark liquid brewed in secrecy and mystery. No one knows the herbs and spells in this liquid but Salako of Wolpi, the head Snake woman of the Moki pueblos. The priests drink of the medicine; in about forty-five seconds it sees the light of day again. They repeat the operation, and so goes on a scene that beggars description. Even scientific equanimity cannot observe without qualms that this is a purification ceremony, carried out by the priests with the ruthlessness of devotion. This feature of the dance, however, will never become popular. Various explanations of the purpose of the medicine have been current. It has been supposed, among others, to be the antidote for the venom of the rattlesnake. Probably it is only for ceremonial purification; at any rate it is a good preparation for the great feast following the dance.

For this feast come bearing trays fair maidens and trim women of gala bread, well cooked meat, corn pudding and other dainties and substantials in profusion. That night there is feasting and every Moki gets what the cowboys call a "mortal gorge." Next day and the day following the boys and girls have great sport in the pueblos. A young man will take a ribbon, a piece of

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The New International Encyclopædia/Hasenclever, Johann Peter

HASENCLEVER, hä'zen-klā'vẽr, (1810-53). A German painter, born at Remscheid. He studied under Schadow, at the Düsseldorf Academy, and then spent some years in Munich and Italy. After 1842 he settled in Düsseldorf. His works are usually humorous genre subjects, such as “A Family Quarrel” (1837, New Pinakothek, Munich); “Tasting Wine,” and “The Reading Room” (1843, both in National Gallery, Berlin); “Topers in the Wine-cellar” (1847, Ravené Gallery, Berlin). He also executed a series of scenes from Kortum's Jobsiade, including “The Examination” (1840, New Pinakothek, Munich); “The School” (1846, Ravené Gallery, Berlin); and “Jobs as Night Watchman” (1852, ib.). His “Gaming-Table” is his most important work of a serious nature. He was also successful in portraiture.


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HE nineteenth century, which prided itself upon the invention of steam and evolution, might have derived a more legitimate title to fame from the discovery of pure mathematics. This science, like most others, was baptised long before it was born; and thus we find writers before the nineteenth century alluding to what they called pure mathematics. But if they had been asked what this subject was, they would only have been able to say that it consisted of Arithmetic, Algebra, Geometry, and so on. As to what these studies had in common, and as to what distinguished them from applied mathematics, our ancestors were completely in the dark.

Pure mathematics was discovered by Boole, in a work which he called the Laws of Thought (1854). This work abounds in asseverations that it is not mathematical, the fact being that Boole was too modest to suppose his book the first ever written on mathematics. He was also mistaken in supposing that he was dealing with the laws of thought: the question how people actually think was quite irrelevant to him, and if his book had really contained the laws of thought, it was curious that no one should ever have thought in such a way before. His book was in fact concerned with formal logic, and this is the same thing as mathematics


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Rh Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. Both these points would belong to applied mathematics. We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

As one of the chief triumphs of modern mathematics consists in having discovered what mathematics really is, a few more words on this subject may not be amiss. It is common to start any branch of mathematics—for instance, Geometry—with a certain number of primitive ideas, supposed incapable of definition, and a certain number of primitive propositions or axioms, supposed incapable of proof. Now the fact is that, though there are indefinables and indemonstrables in every branch of applied mathematics, there are none in pure mathematics except such as belong to general logic. Logic, broadly speaking, is distinguished by the fact that its propositions can be put into a form in which they apply to anything whatever. All pure mathematics—Arithmetic, Analysis,


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76 and Geometry—is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic, such as the syllogism and the other rules of inference. And this is no longer a dream or an aspiration. On the contrary, over the greater and more difficult part of the domain of mathematics, it has been already accomplished; in the few remaining cases, there is no special difficulty, and it is now being rapidly achieved. Philosophers have disputed for ages whether such deduction was possible; mathematicians have sat down and made the deduction. For the philosophers there is now nothing left but graceful acknowledgments.

The subject of formal logic, which has thus at last shown itself to be identical with mathematics, was, as every one knows, invented by Aristotle, and formed the chief study (other than theology) of the Middle Ages. But Aristotle never got beyond the syllogism, which is a very small part of the subject, and the schoolmen never got beyond Aristotle. If any proof were required of our superiority to the mediæval doctors, it might be found in this. Throughout the Middle Ages, almost all the best intellects devoted themselves to formal logic, whereas in the nineteenth century only an infinitesimal proportion of the world's thought went into this subject. Nevertheless, in each decade since 1850 more has been done to advance the subject than in the whole period from Aristotle to Leibniz. People have discovered how to make reasoning symbolic, as it is in Algebra, so that deductions are effected by mathematical rules. They have discovered many rules besides the syllogism, and a new branch of logic, called the Logic of Relatives, has been invented to deal with topics that wholly surpassed the powers of


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Rh the old logic, though they form the chief contents of mathematics.

It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently sceptical to doubt whether it can be proved. And the same holds in other cases where self-evident things are to be proved.

But the proof of self-evident propositions may seem, to the uninitiated, a somewhat frivolous occupation. To this we might reply that it is often by no means self-evident that one obvious proposition follows from another obvious proposition; so that we are really discovering new truths when we prove what is evident by a method which is not evident. But a more interesting retort is, that since people have tried to prove obvious propositions, they have found that many of them are false.


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78 is often a mere will-o'-the-wisp, which is sure to lead us astray if we take it as our guide. For instance, nothing is plainer than that a whole always has more terms than a part, or that a number is increased by adding one to it. But these propositions are now known to be usually false. Most numbers are infinite, and if a number is infinite you may add ones to it as long as you like without disturbing it in the least. One of the merits of a proof is that it instils a certain doubt as to the result proved; and when what is obvious can be proved in some cases, but not in others, it becomes possible to suppose that in these other cases it is false.

The great master of the art of formal reasoning, among the men of our own day, is an Italian, Professor Peano, of the University of Turin. He has reduced the greater part of mathematics (and he or his followers will, in time, have reduced the whole) to strict symbolic form, in which there are no words at all. In the ordinary mathematical books, there are no doubt fewer words than most readers would wish. Still, little phrases occur, such as therefore, let us assume, consider, or hence it follows. All these, however, are a concession, and are swept away by Professor Peano. For instance, if we wish to learn the whole of Arithmetic, Algebra, the Calculus, and indeed all that is usually called pure mathematics (except Geometry), we must start with a dictionary of three words. One symbol stands for zero, another for number, and a third for next after. What these ideas mean, it is necessary to know if you wish to become an arithmetician. But after symbols have been invented for these three ideas, not another word is required in the whole development. All future symbols are symbolically explained by means of these


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Rh three. Even these three can be explained by means of the notions of relation and class; but this requires the Logic of Relations, which Professor Peano has never taken up. It must be admitted that what a mathematician has to know to begin with is not much. There are at most a dozen notions out of which all the notions in all pure mathematics (including Geometry) are compounded. Professor Peano, who is assisted by a very able school of young Italian disciples, has shown how this may be done; and although the method which he has invented is capable of being carried a good deal further than he has carried it, the honour of the pioneer must belong to him.

Two hundred years ago, Leibniz foresaw the science which Peano has perfected, and endeavoured to create it. [sic]He was prevented from succeeding by respect for the authority of Aristotle, whom he could not believe guilty of definite, formal fallacies; but the subject which he desired to create now exists, in spite of the patronising contempt with which his schemes have been treated by all superior persons. From this "Universal Characteristic," as he called it, he hoped for a solution of all problems, and an end to all disputes. "If controversies were to arise," he says, "there would be no more need of disputation between two philosophers than between two accountants. For it would suffice to take their pens in their hands, to sit down to their desks, and to say to each other (with a friend as witness, if they liked), 'Let us calculate.'" This optimism has now appeared to be somewhat excessive; there still are problems whose solution is doubtful, and disputes which calculation cannot decide. But over an enormous field of what was formerly controversial, Leibniz's dream has become sober fact. In the whole philosophy of mathematics, which


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80 used to be at least as full of doubt as any other part of philosophy, order and certainty have replaced the confusion and hesitation which formerly reigned. Philosophers, of course, have not yet discovered this fact, and continue to write on such subjects in the old way. But mathematicians, at least in Italy, have now the power of treating the principles of mathematics in an exact and masterly manner, by means of which the certainty of mathematics extends also to mathematical philosophy. Hence many of the topics which used to be placed among the great mysteries—for example, the natures of infinity, of continuity, of space, time and motion—are now no longer in any degree open to doubt or discussion. Those who wish to know the nature of these things need only read the works of such men as Peano or Georg Cantor; they will there find exact and indubitable expositions of all these quondam mysteries.

In this capricious world, nothing is more capricious than posthumous fame. One of the most notable examples of posterity's lack of judgment is the Eleatic Zeno. This man, who may be regarded as the founder of the philosophy of infinity, appears in Plato's Parmenides in the privileged position of instructor to Socrates. He invented four arguments, all immeasurably subtle and profound, to prove that motion is impossible, that Achilles can never overtake the tortoise, and that an arrow in flight is really at rest. After being refuted by Aristotle, and by every subsequent philosopher from that day to our own, these arguments were reinstated, and made the basis of a mathematical renaissance, by a German professor, who probably never dreamed of any connection between himself and Zeno. Weierstrass, by strictly


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Rh banishing from mathematics the use of infinitesimals, has at last shown that we live in an unchanging world, and that the arrow in its flight is truly at rest. Zeno's only error lay in inferring (if he did infer) that, because there is no such thing as a state of change, therefore the world is in the same state at any one time as at any other. This is a consequence which by no means follows; and in this respect, the German mathematician is more constructive than the ingenious Greek. Weierstrass has been able, by embodying his views in mathematics, where familiarity with truth eliminates the vulgar prejudices of common sense, to invest Zeno's paradoxes with the respectable air of platitudes; and if the result is less delightful to the lover of reason than Zeno's bold defiance, it is at any rate more calculated to appease the mass of academic mankind.

Zeno was concerned, as a matter of fact, with three problems, each presented by motion, but each more abstract than motion, and capable of a purely arithmetical treatment. These are the problems of the infinitesimal, the infinite, and continuity. To state clearly the difficulties involved, was to accomplish perhaps the hardest part of the philosopher's task. This was done by Zeno. From him to our own day, the finest intellects of each generation in turn attacked the problems, but achieved, broadly speaking, nothing. In our own time, however, three men—Weierstrass, Dedekind, and Cantor—have not merely advanced the three problems, but have completely solved them. The solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty. This achievement is probably the greatest of which our age has to boast; and I know of no age (except perhaps the golden


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82 age of Greece) which has a more convincing proof to offer of the transcendent genius of its great men. Of the three problems, that of the infinitesimal was solved by Weierstrass; the solution of the other two was begun by Dedekind, and definitively accomplished by Cantor.

The infinitesimal played formerly a great part in mathematics. It was introduced by the Greeks, who regarded a circle as differing infinitesimally from a polygon with a very large number of very small equal sides. It gradually grew in importance, until, when Leibniz invented the Infinitesimal Calculus, it seemed to become the fundamental notion of all higher mathematics. Carlyle tells, in his Frederick the Great, how Leibniz used to discourse to Queen Sophia Charlotte of Prussia concerning the infinitely little, and how she would reply that on that subject she needed no instruction—the behaviour of courtiers had made her thoroughly familiar with it. But philosophers and mathematicians—who for the most part had less acquaintance with courts—continued to discuss this topic, though without making any advance. The Calculus required continuity, and continuity was supposed to require the infinitely little; but nobody could discover what the infinitely little might be. It was plainly not quite zero, because a sufficiently large number of infinitesimals, added together, were seen to make up a finite whole. But nobody could point out any fraction which was not zero, and yet not finite. Thus there was a deadlock. But at last Weierstrass discovered that the infinitesimal was not needed at all, and that everything could be accomplished without it. Thus there was no longer any need to suppose that there was such a thing. Nowadays, therefore, mathematicians are more dignified than Leibniz: instead of talking about the infinitely small, they talk about the infinitely great—a subject


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The New International Encyclopædia/Hasenclever, Wilhelm

HASENCLEVER, (1837-89). A German socialist agitator, born in Arnsberg, Westphalia. He received his education at the gymnasium of his native city, and then became a tanner. In 1862 he was chosen editor of the Westfälische Volkszeitung, in 1871 president of the Allgemeiner Deutscher Arbeiterverein, and in 1875 presiding officer of the newly organized Socialist-Labor Party of Germany. He was elected to the Reichstag several times, and controlled or edited various socialist newspapers.


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Rh which, however appropriate to monarchs, seems, unfortunately, to interest them even less than the infinitely little interested the monarchs to whom Leibniz discoursed.

The banishment of the infinitesimal has all sorts of odd consequences, to which one has to become gradually accustomed. For example, there is no such thing as the next moment. The interval between one moment and the next would have to be infinitesimal, since, if we take two moments with a finite interval between them, there are always other moments in the interval. Thus if there are to be no infinitesimals, no two moments are quite consecutive, but there are always other moments between any two. Hence there must be an infinite number of moments between any two; because if there were a finite number one would be nearest the first of the two moments, and therefore next to it. This might be thought to be a difficulty; but, as a matter of fact, it is here that the philosophy of the infinite comes in, and makes all straight.

The same sort of thing happens in space. If any piece of matter be cut in two, and then each part be halved, and so on, the bits will become smaller and smaller, and can theoretically be made as small as we please. However small they may be, they can still be cut up and made smaller still. But they will always have some finite size, however small they may be. We never reach the infinitesimal in this way, and no finite number of divisions will bring us to points. Nevertheless there are points, only these are not to be reached by successive divisions. Here again, the philosophy of the infinite shows us how this is possible, and why points are not infinitesimal lengths.

As regards motion and change, we get similarly curious results. People used to think that when a thing changes, it must be in a state of change, and that when a thing


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 Arabia. Zwemer says the fertility of the highland region of J. Akhdar is wonderful and is in striking contrast to the barrenness of so much of the coast; water issues in perennial springs from many rocky clefts, and is carefully husbanded by the ingenuity of the people; underground channels, known here as faluj, precisely similar to the kanat or karez of Persia and Afghanistan, are also largely used. The principal villages on the eastern slopes are Rustak, Nakhl and Semail in the well-watered valley of the same name; on the western slopes are Tanuf and Nizwa, lying immediately below the highest summit of the range; Semed, Ibra and Bidiya in the W. Betha are all well-built villages with palm-groves and irrigated fields. In the north-west the Dhahira district sloping towards the Jewasimi coast is more steppe-like in character; but there two oases of great fertility are found, of which Birema, visited by both Miles and Zwemer, supports a population of 15,000. West of Abu Dhabi a low flat steppe with no settled inhabitants extends up to the Katr peninsula, merging on the north into the saline marshes which border the Persian Gulf, and on the south into the desert.

The great desert known as the Dahna or the Rubʿa el Khali (“the empty quarter”) is believed to cover all the interior of southern Arabia from the borders of Yemen in the west to those of Oman in the east. Halévy in Nejran, Von Wrede in Hadramut, and Wellsted in Oman reached its edge, though none of them actually entered it, and the guides accompanying them all concurred in describing it as uninhabitable and uncrossed by any track. Its northern fringe is no doubt frequented by the Bedouin tribes of southern Nejd after the rains, when its sands, like those of the northern desert, produce herbage; but towards the east, according to Burckhardt’s information, it is quite without vegetation even in the winter and spring. The farthest habitable spot to the south of Nejd is the Wadi Yabrin, which L. Pelly heard of from the Ahl Murra Bedouins as once a fertile district, and which still produces dates, though, owing to malaria, it is now deserted; thence southward to the Hadramut valley no communication is known to exist.

[Geology.—The geological structure of Arabia is very similar to that of Egypt. The oldest rocks consist of granite and schist, penetrated by intrusive dykes, and upon this foundation rest the flat-lying sedimentary deposits, beginning with a sandstone like the Nubian sandstone of Egypt. In the northern part of Arabia the crystalline rocks form a broad area extending from the peninsula of Sinai eastwards to Hail and southwards at least as far as Mecca. Towards the north the crystalline floor is overlaid by the great sandstone series which covers nearly the whole of the country north of Hail. Upon the sandstone rest a few scattered outliers of limestone, probably of Cretaceous age, the largest of which occur near Jauf and east of Bureda. Over both sandstone and granite great sheets of lava have been poured, and these, protecting the softer beds beneath from further denudation, now stand up as the high plateaus and hills called harra. Volcanic cones still exist in large numbers, and the sheets of lava appear as fresh as any recent flows of Etna or Vesuvius. Arabian manuscripts describe an eruption on the harra near Medina in 1256. In the south of Arabia the crystalline floor appears at intervals along the southern coast and on the shores of the Gulf of Oman. At Marbat the granite is overlaid by sandstone, presumably the Nubian sandstone: this is followed by marls containing Cenomanian fossils; and these are overlaid by Upper Cretaceous limestones, upon which rest isolated patches of Alveolina limestone. Generally, however, the Cretaceous beds do not appear, and the greater part of southern Arabia seems to be formed of Alveolina and nummulite limestones of Tertiary age. An extinct volcano occurs at Aden, and volcanic rocks are found at other places near the Straits of Bab-el-Mandeb. Throughout the whole of Arabia, so far as is known, the sedimentary beds show no signs of any but the most gentle folding. Faulting, however, is by no means absent, and some of the faults are of considerable magnitude. The Gulf of Akaba is a strip of country which has been let down between two parallel faults, and several similar faulted troughs occur in the Sinai peninsula. The Red Sea itself is a great trough bounded by faults along each side.]

Climate.—Owing to its low latitude and generally arid surface, Arabia is on the whole one of the hottest regions of the earth; this is especially the case along the coasts of the Persian Gulf and the southern half of the Red Sea, where the moist heat throughout the year is almost intolerable to Europeans. In the interior of northern and central Arabia, however, where the average level of the country exceeds 3000 ft., the fiery heat of the summer days is followed by cool nights, and the winter climate is fresh and invigorating; while in the highlands of Asir and Yemen in the south-west, and of Oman in the east, the summer heat is never excessive, and the winters are, comparatively speaking, cold.

In the northern desert the temperature is subject to extreme variations. Nolde states that on the 1st of February 1893 in the desert north of Hail the thermometer fell from 78° a little before sunset to 18° a quarter of an hour after. The midday temperatures recorded by Huber at Hail during January and the first half of February average about 65° F., and water froze on several nights; at Medina the winters are cold and night frosts of frequent occurrence, and these conditions prevail over all the western part of the Nejd plateau. In the east where the elevation is lower the climate is warmer. In the elevated highland district which extends from Taif to within 50 m. of Aden, the summer heat is tempered by the monsoon winds, and the seasonal variation of temperature is less marked. From observations made at Sana by Manzoni, Deflers and Glaser, the mean temperature for the year of that city at an altitude of 7300 ft. and in 15° 22′ N. appears to be 60° F.; for July the mean maximum was 77°, mean minimum 54°; for January the figures were 62° and 40° respectively, the lowest recorded temperature in 1878 was 26.6° on the 26th of January. At Aden at the sea-level the mean temperature for the year is 83°; the highest observed temperature in 1904 was 97.3°, the lowest 67.4°.

The rainfall throughout northern and central Arabia is chiefly in the winter months between October and April, and is scanty and irregular. Doughty states that in 1876 rain to wet the ground had not fallen for three years at Medain Salih; in that year showers fell on the 29th of December and on two days in January and again in March. After a very hot summer the bright weather changed to clouded skies on the 2nd of October, rain fell tempestuously the same evening, and there were showery days and nights till the 14th. The autumn rains fell that year abundantly in the Nafud towards Jauf, but very little in the basin of the W. Hamd (on the western slope). Doughty adds that the Nejd highlands between Kasim and Mecca are watered yearly by seasonable rains, which at Taif are expected about the end of August and last commonly from four to six weeks. This appears to be about the northern limit reached by the south-west monsoon, which from June to September brings a fairly abundant rainfall to the Yemen highlands, though the Tehama remains almost entirely rainless. The rainfall is heaviest along the western fringe of the plateau, and penetrates inland in decreasing quantity over a zone which perhaps extends to 100 m. in width. In good seasons it is sufficient for the cultivation of the summer crop of millet, and for the supply of the perennial streams and springs, on which the irrigation of the winter crops of wheat and barley depend. The amount measured at Dhala at the extreme south of the plateau at an elevation of 4800 ft. was in 1902 as follows:—June, 4.0 in.; July, 5.5; August, 5.8; September, 1.9. Only slight showers were recorded in the other months of the year. At higher elevations the rainfall is no doubt heavier; Manzoni mentions that at Sana there was constant rain throughout August and September 1878, and that the thermometer during August did not reach 65°. In the Tehama occasional showers fall during the winter months; at Aden the average rainfall for the year is 2.97 in., but during 1904 only 0.5 in. was recorded. Snow falls on the Harra and on the Tehama range in northern Arabia, and Nolde records a fall of snow which lay on the Nafud on the 1st of February 1893. It also falls on J. Akhdar in Oman, but is very rarely known on the Yemen mountains, probably because the precipitation during the winter months is so slight.

The prevailing winds in northern Arabia as far as is known are from the west; along the southern coast they are from the east; at Sana there is generally a light breeze from the north-north-west from 9 to 11 a.m., from noon till 4 p.m. a steady and often strong wind blows from the south-south-east, which dies away later. The climate is extremely dry, but this is compensated for by the heavy mists which sweep up from the plains during the rainless months and exercise a most beneficial effect in the coffee-growing districts. This phenomenon is known as the sukhemani or amama. In the morning the Tehama, as seen from the mountain tops, appears buried in a sea of white cloud; towards noon the clouds drift up the mountain slopes and cover the summits with wreaths of light mist charged with moisture which condenses on the trees and vegetation; in the afternoon they disappear, and the evenings are generally clear and still.

Fauna.—The wild animals of Arabia are all of the desert-loving type: antelopes and gazelles are found in small numbers throughout the peninsula; the latter are similar to the chikara or ravine deer of India. The larger antelopes, so common on the African side of the Gulf of Aden, are not found, except one variety, the Oryx beatrix (called by the Arabs, wild cow), which is an inhabitant of the Nafud between Tema and Hail; it is about the size of a donkey, white, and with long straight horns. Hares are numerous both in the desert and in cultivated tracts. In the Yemen mountains the wal, a wild goat with massive horns, similar to the Kashmir ibex, is found; monkeys also abound. Among smaller animals the jerboa and other descriptions of rat, and the wabar or cony are common; lizards and snakes are numerous, most of the latter being venomous. Hyenas, wolves and panthers are found in most parts of the country, and in the mountains the leopard and wild cat. Of birds the ostrich is found in the Nafud and in the W. Dawasir. Among game birds the bustard, guinea fowl, sand grouse (kata), blue rock, green pigeon, partridge, including a large chikor (akb) and a small species similar to the Punjab sisi; quail and several kinds of duck and snipe are met with. In the cultivated parts of Yemen and Tehama small birds are very numerous, so also are birds of prey, vultures, kites and hawks.

Insects of all sorts abound; scorpions, centipedes, spiders, and an ugly but harmless millipede known in Yemen as hablub are very common in summer. Ants and beetles too are very numerous, and anthills are prominent features in many places. Locusts appear in great swarms and do much damage; fires are lighted at night

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84 moves, it is in a state of motion. This is now known to be a mistake. When a body moves, all that can be said is that it is in one place at one time and in another at another. We must not say that it will be in a neighbouring place at the next instant, since there is no next instant. Philosophers often tell us that when a body is in motion, it changes its position within the instant. To this view Zeno long ago made the fatal retort that every body always is where it is; but a retort so simple and brief was not of the kind to which philosophers are accustomed to give weight, and they have continued down to our own day to repeat the same phrases which roused the Eleatic's destructive ardour. It was only recently that it became possible to explain motion in detail in accordance with Zeno's platitude, and in opposition to the philosopher's paradox. We may now at last indulge the comfortable belief that a body in motion is just as truly where it is as a body at rest. Motion consists merely in the fact that bodies are sometimes in one place and sometimes in another, and that they are at intermediate places at intermediate times. Only those who have waded through the quagmire of philosophic speculation on this subject can realise what a liberation from antique prejudices is involved in this simple and straightforward commonplace.

The philosophy of the infinitesimal, as we have just seen, is mainly negative. People used to believe in it, and now they have found out their mistake. The philosophy of the infinite, on the other hand, is wholly positive. It was formerly supposed that infinite numbers, and the mathematical infinite generally, were self-contradictory. But as it was obvious that there were infinities—for example, the number of numbers—the contradictions of infinity seemed unavoidable, and philosophy seemed to


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Rh have wandered into a "cul-de-sac." This difficulty led to Kant's antinomies, and hence, more or less indirectly, to much of Hegel's dialectic method. Almost all current philosophy is upset by the fact (of which very few philosophers are as yet aware) that all the ancient and respectable contradictions in the notion of the infinite have been once for all disposed of. The method by which this has been done is most interesting and instructive. In the first place, though people had talked glibly about infinity ever since the beginnings of Greek thought, nobody had ever thought of asking, What is infinity? If any philosopher had been asked for a definition of infinity, he might have produced some unintelligible rigmarole, but he would certainly not have been able to give a definition that had any meaning at all. Twenty years ago, roughly speaking, Dedekind and Cantor asked this question, and, what is more remarkable, they answered it. They found, that is to say, a perfectly precise definition of an infinite number or an infinite collection of things. This was the first and perhaps the greatest step. It then remained to examine the supposed contradictions in this notion. Here Cantor proceeded in the only proper way. He took pairs of contradictory propositions, in which both sides of the contradiction would be usually regarded as demonstrable, and he strictly examined the supposed proofs. He found that all proofs adverse to infinity involved a certain principle, at first sight obviously true, but destructive, in its consequences, of almost all mathematics. The proofs favourable to infinity, on the other hand, involved no principle that had evil consequences. It thus appeared that common sense had allowed itself to be taken in by a specious maxim, and that, when once this maxim was rejected, all went well.

The maxim in question is, that if one collection is part


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86 of another, the one which is a part has fewer terms than the one of which it is a part. This maxim is true of finite numbers. For example, Englishmen are only some among Europeans, and there are fewer Englishmen than Europeans. But when we come to infinite numbers, this is no longer true. This breakdown of the maxim gives us the precise definition of infinity. A collection of terms is infinite when it contains as parts other collections which have just as many terms as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there are an infinite number of terms in the collection. For example, there are just as many even numbers as there are numbers altogether, since every number can be doubled. This may be seen by putting odd and even numbers together in one row, and even numbers alone in a row below:—

There are obviously just as many numbers in the row below as in the row above, because there is one below for each one above. This property, which was formerly thought to be a contradiction, is now transformed into a harmless definition of infinity, and shows, in the above case, that the number of finite numbers is infinite.

But the uninitiated may wonder how it is possible to deal with a number which cannot be counted. It is impossible to count up all the numbers, one by one, because, however many we may count, there are always more to follow. The fact is that counting is a very vulgar and elementary way of finding out how many terms there are in a collection. And in any case, counting gives us what mathematicians call the ordinal number of our terms; that is to say, it arranges our terms in an order or


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The New International Encyclopædia/Hasenpflug, Karl

HASENPFLUG, hä'zen-pflōōG, (1802-58). A German architectural painter, born in Berlin, the son of a shoemaker. After a hard struggle in early life, he became the pupil of the decorative painter Karl Gropius, and a few years later was enabled by a royal stipend to study at the academy, but really owed his development mostly to his own efforts. Attracted by the mediæval architecture of Halberstadt, he made his home there in 1830, and, besides cathedrals, painted chiefly views of ruined castles and convents half buried in snow, which appealed to a large public by their great poetic charm. Especially noteworthy among his works are two views of Cologne Cathedral (1832-33); Erfurt Cathedral (1827); and three views of and in Halberstadt Cathedral (1828-36, the last four in National Gallery, Berlin); “Cloister in Winter” (1840, Kunsthalle, Hamburg); “Mediæval Castle Yard” (1842, Schwerin Gallery); and “Ruined Chapel” (Stettin Museum).


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Rh series, and its result tells us what type of series results from this arrangement. In other words, it is impossible to count things without counting some first and others afterwards, so that counting always has to do with order. Now when there are only a finite number of terms, we can count them in any order we like; but when there are an infinite number, what corresponds to counting will give us quite different results according to the way in which we carry out the operation. Thus the ordinal number, which results from what, in a general sense may be called counting, depends not only upon how many terms we have, but also (where the number of terms is infinite) upon the way in which the terms are arranged.

The fundamental infinite numbers are not ordinal, but are what is called cardinal. They are not obtained by putting our terms in order and counting them, but by a different method, which tells us, to begin with, whether two collections have the same number of terms, or, if not, which is the greater. It does not tell us, in the way in which counting does, what number of terms a collection has; but if we define a number as the number of terms in such and such a collection, then this method enables us to discover whether some other collection that may be mentioned has more or fewer terms. An illustration will show how this is done. If there existed some country in which, for one reason or another, it was impossible to take a census, but in which it was known that every man had a wife and every woman a husband, then (provided polygamy was not a national institution) we should know, without counting, that there were exactly as many men as there were women in that country, neither more nor


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88 less. This method can be applied generally. If there is some relation which, like marriage, connects the things in one collection each with one of the things in another collection, and vice versa, then the two collections have the same number of terms. This was the way in which we found that there are as many even numbers as there are numbers. Every number can be doubled, and every even number can be halved, and each process gives just one number corresponding to the one that is doubled or halved. And in this way we can find any number of collections each of which has just as many terms as there are finite numbers. If every term of a collection can be hooked on to a number, and all the finite numbers are used once, and only once, in the process, then our collection must have just as many terms as there are finite numbers. This is the general method by which the numbers of infinite collections are defined.

But it must not be supposed that all infinite numbers are equal. On the contrary, there are infinitely more infinite numbers than finite ones. There are more ways of arranging the finite numbers in different types of series than there are finite numbers. There are probably more points in space and more moments in time than there are finite numbers. There are exactly as many fractions as whole numbers, although there are an infinite number of fractions between any two whole numbers. But there are more irrational numbers than there are whole numbers or fractions. There are probably exactly as many points in space as there are irrational numbers, and exactly as many points on a line a millionth of an inch long as in the whole of infinite space, There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because,


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Rh if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work.

We can now understand why Zeno believed that Achilles cannot overtake the tortoise and why as a matter of fact he can overtake it. We shall see that all the people who disagreed with Zeno had no right to do so, because they all accepted premises from which his conclusion followed. The argument is this: Let Achilles and the tortoise start along a road at the same time, the tortoise (as is only fair) being allowed a handicap. Let Achilles go twice as fast as the tortoise, or ten times or a hundred times as fast. Then he will never reach the tortoise. For at every moment the tortoise is somewhere and Achilles is somewhere; and neither is ever twice in the same place while the race is going on. Thus the tortoise goes to just as many places as Achilles does, because each is in one place at one moment, and in another at any other moment. But if Achilles were to catch up with the tortoise, the places where the tortoise would have been would be only part of the places where Achilles would have been. Here, we must suppose, Zeno appealed to the maxim that the whole has more terms that the part. Thus if Achilles were


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90 to overtake the tortoise, he would have been in more places than the tortoise; but we saw that he must, in any period, be in exactly as many places as the tortoise. Hence we infer that he can never catch the tortoise. This argument is strictly correct, if we allow the axiom that the whole has more terms than the part. As the conclusion is absurd, the axiom must be rejected, and then all goes well. But there is no good word to be said for the philosophers of the past two thousand years and more, who have all allowed the axiom and denied the conclusion.

The retention of this axiom leads to absolute contradictions, while its rejection leads only to oddities. Some of these oddities, it must be confessed, are very odd. One of them, which I call the paradox of Tristram Shandy, is the converse of the Achilles, and shows that the tortoise, if you give him time, will go just as far as Achilles. Tristram Shandy, as we know, employed two years in chronicling the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that, as years went by, he would be farther and farther from the end of his history. Now I maintain that, if he had lived for ever, and had not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten. For consider: the hundredth day will be described in the hundredth year, the thousandth in the thousandth year, and so on. Whatever day we may choose as so far on that he cannot hope to reach it, that day will be described in the corresponding year. Thus any day that may be mentioned will be written up sooner or later, and therefore no part of the biography will remain permanently unwritten. This paradoxical but perfectly true proposition depends upon the fact


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Rh that the number of days in all time is no greater than the number of years.

Thus on the subject of infinity it is impossible to avoid conclusions which at first sight appear paradoxical, and this is the reason why so many philosophers have supposed that there were inherent contradictions in the infinite. But a little practice enables one to grasp the true principles of Cantor's doctrine, and to acquire new and better instincts as to the true and the false. The oddities then become no odder than the people at the antipodes, who used to be thought impossible because they would find it so inconvenient to stand on their heads.

The solution of the problems concerning infinity has enabled Cantor to solve also the problems of continuity. Of this, as of infinity, he has given a perfectly precise definition, and has shown that there are no contradictions in the notion so defined. But this subject is so technical that it is impossible to give any account of it here.

The notion of continuity depends upon that of order, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry, while order more and more reigns supreme. The investigation of different kinds of series and their relations is now a very large part of mathematics, and it has been found that this investigation can be conducted without any reference to quantity, and, for the most part, without any reference to number. All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives.


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92 The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite differently, and the series which it limits may not approximate to it at all. This improvement also is due to Cantor, and it is one which has revolutionised mathematics. Only order is now relevant to limits. Thus, for instance, the smallest of the infinite integers is the limit of the finite integers, though all finite integers are at an infinite distance from it. The study of different types of series is a general subject of which the study of ordinal numbers (mentioned above) is a special and very interesting branch. But the unavoidable technicalities of this subject render it impossible to explain to any but professed mathematicians.

Geometry, like Arithmetic, has been subsumed, in recent times, under the general study of order. It was formerly supposed that Geometry was the study of the nature of the space in which we live, and accordingly it was urged, by those who held that what exists can only be known empirically, that Geometry should really be regarded as belonging to applied mathematics. But it has gradually appeared, by the increase of non-Euclidean systems, that Geometry throws no more light upon the nature of space than Arithmetic throws upon the population of the United States. Geometry is a whole collection of deductive sciences based on a corresponding collection of sets of axioms. One set of axioms is Euclid's; other equally good sets of axioms lead to other results. Whether Euclid's axioms are true, is a question as to which the the pure mathematician is indifferent; and, what is more, it is a question which it is theoretically impossible to answer with certainty in the affirmative. It might


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Rh be shown, by very careful measurements, that Euclid's axioms are false; but no measurements could ever assure us (owing to the errors of observation) that they are exactly true. Thus the geometer leaves to the man of science to decide, as best he may, what axioms are most nearly true in the actual world. The geometer takes any set of axioms that seem interesting, and deduces their consequences. What defines Geometry, in this sense, is that the axioms must give rise to a series of more than one dimension. And it is thus that Geometry becomes a department in the study of order.

In Geometry, as in other parts of mathematics, Peano and his disciples have done work of the very greatest merit as regards principles. Formerly, it was held by philosophers and mathematicians alike that the proofs in Geometry depended on the figure; nowadays, this is known to be false. In the best books there are no figures at all. The reasoning proceeds by the strict rules of formal logic from a set of axioms laid down to begin with. If a figure is used, all sorts of things seem obviously to follow, which no formal reasoning can prove from the explicit axioms, and which, as a matter of fact, are only accepted because they are obvious. By banishing the figure, it becomes possible to discover all the axioms that are needed; and in this way all sorts of possibilities, which would have otherwise remained undetected, are brought to light.

One great advance, from the point of view of correctness, has been made by introducing points as they are required, and not starting, as was formerly done, by assuming the whole of space. This method is due partly to Peano, partly to another Italian named Fano. To those unaccustomed to it, it has an air of somewhat wilful pedantry. In this way, we begin with the following