Cusp

“ Hence, in the case of a curve which has δ nodes and κ cusps, the apparent diminution 2 (m2 − m − 6) (2δ + 3κ) is too great, and it has in fact to be diminished by 2{2δ (δ − 1) + 6δκ + 9⁄2κ (κ − 1) }, or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps. We have thus finally an expression for 2τ, = m (m − 2) (m2 − 9) − &c.; or dividing the whole by 2, we have the expression for τ given by the third of Plücker’s equations. ”

“ Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also δ and κ nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersections is = m2 − 2δ − 3κ. ”

“ The dentition is very peculiar and characteristic: there are in all thirty-two teeth, tipped with deep crimson; of which twelve belong to the lower jaw; of the remaining twenty ten occupy each side of the upper jaw, and of these the first three are incisors. The first incisor is large, with a long anterior canine-like cusp and a small posterior one; then follow two small single-cusped teeth; which are succeeded by three similar progressively smaller teeth, the first being a canine and the other two premolars ”

“ In the annexed figure, there are shown various examples of the curves named above, when the radii of the rolling and fixed circles are in the ratio of 1 to 3. Since the circumference of a circle is proportional to its radius, it follows that if the ratio of the radii be commensurable, the curve will consist of a finite number of cusps, and ultimately return into itself. In the particular case when the radii are in the ratio of 1 to 3 the epicycloid (curve a) will consist of three cusps external to the circle and placed at equal distances along its circumference. ”

“ All the cusps have been touched by A; all the crosses below except the middle one tend toward the loadstone, not straight, but obliquely; because the pole is in the middle of the plane which before was the plane of the æquator. All cusps touched by places distant from the pole move toward the pole (exactly the same as if they had been rubbed upon the pole itself) , not toward the place where they were rubbed, wherever that may have been in the undivided stone in some latitude between the pole and the æquator. ”

“ I estimated the vacancy between the cusps of the lunes to be 1/4 diameter of the largest.
On measuring the diameters and distance under the microscope, it appeared that the largest was ,0312; a quarter of which is ,0078. The distance of the globules from each other measured ,0111. The difference in the estimation ,0033 is less than 1/300 part of an inch.
The experiment was repeated with a change of the distance of the globules from each other. They were then estimated to be less than the diameter of the large one asunder, but full that of the small one. ”

“ We have herein a better principle of classification; considering cubic curves, in the first instance, according to singularities, the curves are non-singular, nodal (viz. crunodal or acnodal) , or cuspidal; and we see further that there are two kinds of non-singular curves, the complex and the simplex. There is thus a complete division into the five kinds, the complex, simplex, crunodal, acnodal and cuspidal. Each singular kind presents itself as a limit separating two kinds of inferior singularity ”

“ Besides which, the cubit being the unit, the half cubit might serve the purpose of the foot; but the metre, divisible only by two and ten, gave no measure practically corresponding to the foot whatever. It appears to have been considered that decimal arithmetic, although affording great facilities for the computation of numbers, is not equally well suited for the division of material substances. A glance of the eye is sufficient to divide material substances into successive halves, fourths, eighths, and sixteenths. A slight attention will give thirds, sixths, and twelfths. ”

“ The same exactitude is found in the applications of differential calculus to geometry and mechanics. If we imagine a curve cut by a secant at two adjacent points, naming E the interval of the ordinates of these two points, E will be the increment of the abscissa from the first to the second ordinate. It is easy to see that the corresponding increment of the ordinate will be the product of E by the first ordinate divided by its subsecant; augmenting then in this equation of the curve the first ordinate by this increment, we shall have the equation relative to the second ordinate. ”

“ It is mathematically equal to the product of the length of the circumference, multiplied by half the radius. To square a circle of a given diameter in mètres, is the same as giving the number of squares, of a mètre in each side, of which the surface is the equivalent. If, the diameter being given, the exact circumference were known by a sort of inspiration, the superficial extent of the circular space would be deducible from the two numbers, by the mere multiplication of the numerical length of the circumference by the fourth of the diameter, or half the radius. ”

“ Now Obzr being a quadrilateral inscribed in the circle described on bO, we have (by the prop. before cited) bO · zr + Or · bz = br · Oz; and hence bO · zr = br · Oz – Or · bz. But br = sine arc bE = sine arc BD; and since BD is the complement of AB, br = cosAB. In like manner Oz = cosBC, Or = sinAB, and bz = sinBC; hence by substitution, bO · zr = cosAB · cosBC – sinAB · sinBC. By using the same notation as before, we have cos (A + B) = cosA · cosB – sinA · sinB r = (if r = 1) cosA · cosB – sinA · sinB, which is the known formula for the cosine of the sum of two arcs. ”

“ Advantages of the Metric System.—First, it is a decimal system; second, the same form and prefixes are used in every table; third, the standards of length (meter) , volume (liter) , and mass (kilogram) bear a simple relation [Pg 11] to one another. This simple relation between the three standard units may be given thus: first, the liter is a cubic decimeter, and second, the kilogram is the mass of a liter of water. (See Fig. 5) Since the liter is a cubic decimeter, the length of one side is 10 cm. The liter therefore holds 1000 ccm. (10 × 10 × 10) . Therefore, 1 liter = 1 cu. ”

“ We shall not here attempt to explain how the other measures—itinerary, agrarian, of capacity, of weight—were derived from the cubit; to call attention to the traces left in our nomenclature by the duodecimal or sexagesimal system of the Babylonians, even after the complete triumph of the decimal system, is sufficient for our purposes. It is used for instance in the division of the circle into degrees, minutes, and seconds, in the division of the year into months, and of the day into hours and their fractions. ”

“ On the other hand, the calculation by means of the curve constants (under consideration of the corrections due to the error of the -scale mentioned on p. 513) :
The values for are
Eventually one can also make a comparison of the different theories, by determining the constants A and B, or C and D themselves, from equations (16) be means of the measured values of M and E and the cathode-ray-value of , and to calculate the curve with their aid. The calculated values of the curve are:
By means of the values, the curve can be calculated according to equation (18) or (21) . ”

“ To fit a curve to the given data, after the constants have been determined it is necessary to find, by solving a cubic equation for each value, the values of t corresponding to the x’s of the respective classes. ”

“ From the unit of linear measure of metre is derived the unit of the measure of capacity or Litre. This represents the cube of one-tenth part of a metre, or a cubic decimetre, and its multiples and sub-divisions with their corresponding equivalents in Imperial fluid measure are as follows:—
The unit of weight in the metric system is the Gramme. ”

“ Now the notations and represent the sine and the cosine of the angle included between and combined in each case with certain other simple notions. But the sine and cosine combined with these auxiliary notions are incomparably more amenable to analytical transformation than the simple sine and cosine of trigonometry, exactly as numerical quantities combined (as in algebra) with the notion of positive or negative quality are incomparably more amenable to analytical transformation than the simple numerical quantities of arithmetic. ”

“ The unit of surface is the are, which is 100 square meters. The theoretical unit of volume is the stere, which is a cubic meter. The unit of volume for the purposes of the market is the liter, which is the volume of one kilogram of distilled water at its maximum density and is intended to be one cubic decimeter. For 10 times, [41] 100 times, 1,000 times and 10,000 times one of these units, the prefixes, deca-, hecto-, kilo- and myria- are used. For 1-10, 1-100 and 1-1,000 of the units, the prefixes deci-, centi- and milli- are used. ”

“ If increases at the rate of 2 inches per second as it passes through the value inches, at what rate must change when inch, in order that the function shall remain constant?
Solution. Let ; then
Substituting in (57a) , or By (33) ,§94 But x = 3, y = 1, ∴ ft. per second. Ans. Let P be the point (x, y, z) on the surface given by the equation
(E) and let PC and AP be sections made by planes through P parallel to the YOZ- and XOZ-planes respectively. Along the curve AP, y is constant; therefore, from (E) , z is an implicit function of x alone, and we have, from (57a) ,
Point P on a surface. ”

“ Thus a auxiliary table is calculated, which gives the -values corresponding to a most tight row of -values, then (when the constants and are known) the corresponding by equation (7.) can be compared to every . It is easily achieved to find approximate values for both "curve constants" and , and then to calculate the improvements still to be made, by the method of least squares.
The curve constants determined in this way, can be compared with those values partly given from the dimensions of the apparatus alone, and partly from the connection with the value found with respect to cathode rays. ”

“ If there are 1, 4, 7, 10, etc., zeros,
then use the middle third of the K scale. If there are no zeros or 3, 6,
9, 12, etc., zeros, then use the right-hand third of the K scale. For
example:
Example 41: cube_root ( 185 ) = 5.70
Since there are 3 digits in the given number, we set the indicator on
185 in the right-hand third of the K scale, and read the result 570 on
the D scale. We can place the decimal point by thinking of the nearest
perfect cube, which is 125. Therefore, the decimal point must be placed
so as to give 5.70, which is nearest to 5, the cube root of 125. ”

“ Indeed, the late Professor Newcomb stated that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." ”

“ We knew beforehand that it had to be a cubic equation. Each cubic equation has three roots for every value of the independent variable, which is in space. It also has three roots for every value of the time. Basically, that means that if any primal unit exists in our space, it exists in three forms, the positive, the negative and the neutral. These units are the positron, the negatron, and the neutron. Those three are the three solutions in space to the co-ordinates of the existential primal point. ”

“ Accordingly the unit of energy will be in the English system, that required to lift one pound through about the thirty-second part of a foot; in terms of metric units, it will be that required to lift a kilogram through about one tenth part of a metre.
Every person is at liberty to measure and record quantities in terms of any unit which he likes. He may use the yard for linear measurement and the litre for cubic measurement, only there will then be a complicated relation between his different results. ”

“ The suffixes are chosen so that we may have ΔΣun = un, whatever n may be; and therefore (§ 4) Σun may be regarded as being the sum of the terms of the series up to and including un-1. Thus if we write Σun-1 = C + un-2, where C is any constant, we shall have
and so on. This is true whatever C may be, so that the knowledge of ... un-1, un, ... gives us no knowledge of the exact value of Σun; in other words, C is an arbitrary constant, the value of which must be supposed to be the same throughout any operations in which we are concerned with values of Σu corresponding to different suffixes. ”

“ What gives the metric system pre-eminence is the simplicity and uniformity of its decimal scale, forming part and parcel as it does of the decimal system of notation, and lending itself to a decimal coinage as in France, Germany, Italy, and Spain. The metre is organically related to all measures of length, surface, capacity, solidity, and weight. A cubic centimetre of water, taken as it melts in a vacuum, at 4° C., the temperature of maximum density, is the gram from which other weights are derived ”

“ It is a very difficult matter to set the compasses so that by commencing at any one point and stepping the compasses around the circle continuously in one direction, the compass point shall fall into the precise point from which it started, for if the compass point be set the 1⁄200th inch out, the last space will come an inch out in a circle having 200 points of divisions. ”

“ Assuming it as the basis of our computation there are in a cubic centimeter of gas under normal conditions molecules and the weight of a hydrogen atom is . These numbers can be made more significant to the ordinary reader with the aid of an illustration. If a million men were to be set counting as fast as they could count, say at the rate of 200 a minute, they could count out the number of molecules in a cubic centimeter in just 252 thousand years if none of them ever stopped to eat, sleep, or die. ”