INVOLUTION 



3030 



IODINE 



Cutworm Polyp 



Daphnia Protozoa 



Earthworm Sea Anemone 



Earwig Sea Cucumbers 



Hairworm Sea Urchin 



Hookworm Sponge 



Hydra Starfish 



Jellyfish Tent Caterpillar 



Leech Trepang 



Lobworm Vermes 



Measuring Worm Worms 



INVOLUTION, involu'shun, in mathemat- 

 ics, is the process of raising a number to any 

 required power. By power is meant the prod- 

 uct obtained by using the number two or 

 more times as a factor. For example, 64, the 

 product obtained by using 4 three times as a 

 factor, is the third power of 4, since 4X4X4=64. 

 The power is indicated by a small figure, 

 called an exponent, which is placed at the right 

 of and a little above the number. Thus: 



=:3 2 = 9, the second power of 3. 

 3X3X3 3 3 = 27, the third power of 3. 

 3x3x3x3rr3" = 81, the fourth power of 3, etc. 



The first power of a number is the number it- 

 self; thus 3 3 1 . When a number is taken 

 twice as a factor it is said to be squared; when 

 taken three times it is cubed. Also the second 

 power of any number is its , square, and the 

 third power its cube. 



When a fraction is raised to any power, both 

 the numerator and denominator must be raised 

 to that power. For example, the fraction % 

 raised to the third power = 



3X3X3_27 

 4X4X4^64' 



The product of any two powers of a number 

 may be found by adding the exponents and 

 raising the number to the power indicated by 

 the sum. Thus, 42X43=42+3, or 45, O r 1,024. 

 Again, when it is required to raise a number 

 with an exponent to any power, such as (3 2 ) 3 , 

 the exponents may be multiplied and the 

 product be considered the exponent of the 

 number to be raised. Thus, (3 2 )3=3 2X 3, =36, 

 =729. Any number, therefore, is raised to the 

 sixth power when its square is cubed; likewise 

 cubing its cube raises it to the ninth power. 



By the application of the following rule, the 

 square of any number composed of two figures 

 may be found: 



The square of any number composed of two 

 figures is equal to the square of the tens, plus 

 twice the product of the tens and the units, plus 

 the square of the units. 



Square 35. 



Process : (35) 2 =30 2 +2(30X5)+5 2 =900+ 

 2 ( 150) +25=900+300+25= 1 ,225. 



A number consisting of two figures may be 

 cubed by applying the following rule: 



The cube of any number consisting of two 

 figures is equal to the cube of the tens, plus three 

 times the product of the tens squared and the 

 units, plus three times the product of the tens 

 and the units squared, plus the units cubed. 



Cube 35. 



Process: (35)3=303+3(30 2 X5)+3(30X5 2 ) + 

 53 = 27,000 + 3(4,500) +3(750) + 125 = 27,000 + 

 13,500+2,250+125=42,875. 



The process of finding the root of a number, 

 the reverse of involution, is known as evolution 

 (which see). B.M.W. 



10, i' o, in Greek mythology, the daughter 

 of the king of Argos, beloved by Zeus, who, 

 to protect her from the jealousy of Hera, 

 changed her into a white heifer. Hera con- 

 trived, however, to obtain the animal from 

 Zeus, and gave her to the care of the hundred- 

 eyed Argus, from whom she was rescued by 

 Hermes. Hera then sent the gadfly to tor- 

 ment the unfortunate lo, and the latter wan- 

 dered all over the world in her attempts to 

 escape from her tormentor. In Egypt she 

 found rest and freedom from persecution. On 

 the banks of the Nile she assumed her original 

 form. Her son Epaphus, later king of Egypt, 

 was the reputed founder of the famous city 

 of Memphis. 



IODINE, i' o din, a heavy, lustrous, bluish- 

 black, solid substance, usually occurring in flat 

 crystals. It has a disagreeable odor resembling 

 that of chlorine, but much milder. When 

 heated it yields the violet vapor to which it 

 owes its name (from the Greek iodes, meaning 

 violctlike). Iodine was discovered in 1812 

 by B. Courtois, a French manufacturer of salt- 

 peter. He found it in kelp, the ashes of sea- 

 weed. The chemists Clement, Gay-Lussac and 

 Sir Humphry Davy examined the new sub- 

 stance and pronounced it an element (see 

 CHEMISTRY). Iodine melts at 237 F. and 

 boils at 363. Its vapor is one of the heaviest 

 gases known, being nearly nine times as heavy 

 as air and so dark in color that a vessel four 

 inches wide filled with it looks black and quite 

 opaque. Pure water will dissolve only one-six- 

 thousandth of its own weight of iodine, but 

 water containing half its weight of potassium 

 iodide will dissolve its own weight of iodine. 

 The solution so obtained is dark brown in color. 

 More dilute solutions are red or yellow. Iodine 

 likewise forms brown solutions in alcohol, ether 

 and glycerine, but in chloroform, carbon di- 

 sulphide, benzine and light petroleum oils 



