in driving back the constant invasions of the Danes, the most terrible warriors of Europe, and a large part of his reign was spent in preserving the liberty of his country against these northern foes. At first he was unsuccessful, and by 878 the invaders had overrun the entire kingdom of the West Saxons, while Alfred was driven into its forests. But he refused to be beaten, and soon the tide of fortune turned. Building a stronghold on an island in the wastes of Somersetshire, still known as Athelney (the island of the nobles), he made frequent sallies against the enemy, and soon found himself at the head of an army with which he totally defeated them. He then built England's first fleet and soon grew so powerful, both by land and sea, that he was recognized as sovereign of all England. During the years of peace which followed Alfred busied himself in rebuilding the cities which had suffered in the wars, in training the people in the use of arms and in founding those wise laws and institutions which helped so much in making England great and happy in later years. In an age of ignorance he was a fine scholar, and did much in founding schools and encouraging literature. Toward the close of his reign, after a hard contest of three years, he was again victorious over his old enemies, the Danes. He died in 901, leaving his country in peace and prosperity as the result of that wise and energetic rule which endeared him to all Englishmen as their best and greatest ruler.
Algæ (Sl'je). One of the great divisions of Thallophytes (the lowest group of plants), being distinguished from the Fungi by containing the green coloring matter known as chlorophyll. This enables them to manufacture their own food and so to live independently of all other organisms. They are of special interest as representing the most primitive forms of the plant kingdom, from which all other groups of plants have probably been derived. They are exclusively water plants, either living in the water or in damp places, and are commonly known as "seaweeds," although they are abundant in fresh as well as in salt water. Their bodies are of various sizes and degrees of complexity. Some are only a single cell and are microscopic in size, while others are very complex and huge in size, as the giant kelps of the ocean. There are four great groups of Algæ, named for their differences in color. The Cyanophyceæ or blue-green algae are the simplest, and are characterized by possessing a blue pigment in addition to the green chlorophyll, which gives them a bluish-green hue. The Chlorophyceæ or "green algæ" have no other pigment than the green chlorophyll. These two groups are characteristic of fresh waters, although they have their marine representatives. The two following groups are characteristic of salt waters, but have representatives in fresh waters. The Phæophyceæ or brown algæ have a yellowish to brown pigment in addition to the chlorophyll, which gives their bodies various shades from olive to yellow and brown. They include the common large and coarser seaweeds cast up by the waves. The Rhodophyceæ or red algae have a red pigment in addition to the chlorophyll, and their graceful and often very delicate bodies, beautifully tinted with various shades of red, are among the most attractive plants of the seashore. For a further account see under the names of the four groups.
Al'gebra is a branch of mathematics which deals chiefly with "functions" or general values instead of special values as in arithmetic. The ancient Egyptians practised simple equations, an example being this: " Its whole added to its seventh gives 19, how much is it?" In other words, they solved the equation . The Greeks added something to algebra; thus Euclid, about 300 B. C., knew that . Other steps in advance were made in Alexandria and Persia. But algebra was only used as a help to arithmetic until Viéte or Vieta, a Frenchman, in 1591 made of it an independent science. As to the usefulness of algebra, it can only be said that it is needful to all advanced work in mathematics. Its value to the professional man or workman may not be great, except that it is well for every man to know a little of each of the branches of truth.
The teaching of algebra might well follow the historical order; and begin with simple equations as did the Egyptians. For here algebra is of obvious use in making the problems of arithmetic more simple. Let one ask the following "catch" question: "A goose weighs six pounds and half its own weight, what is the weight of the goose?" The answer is seldom given rightly without setting x for the weight of the goose, thus: which gives the answer 12 pounds. It is better to begin, however, with practical questions. The most important modern change in the teaching of algebra has been brought about by Professor Chrystal, who has called attention to the nature of general functions as the real object of study in this science. A knowledge of general functions, such as the following for a quadratic equation, , has always been implied in the teaching of algebra; but it has only lately been insisted upon. It has been usual to teach the use of root signs and signs for brackets as if they formed a part of algebra; but in reality these operations belong to pure arithmetic.
