DICTIONARY

OF

SCIENCE, LITERATURE, AND ART.

POLE-AXE

POLES AND POLARS Poles and Polars. The locus of the that point; indeed, generally, the polars of a harmonic centres of the (n-r)th order, taken point o situated on the primitive curve Cn, all with respect to a point or pole o, of the n inter- touch the latter in o, the common tangent being, sections of a given curve C by a transversal of course, the polar line of o. The first polar which constantly passes through o, is called the C-1 of any point o is the locus of the points 4th polar of o with respect to the given (primitive) of contact of all tangents drawn from o to the curve C. It is itself a curve of the (n-r)th primitive curve; so that n (n-1), or the nunorder [HARMONIC CENTRES], and possesses very ber of intersections of C, and C-1, is the important properties. Each point in the plane, maximum, and in general the actual number of therefore, has (n-1) distinct polars of the such tangents; this number, therefore, indicates orders n-1, n-2, 2, 1, respectively. the class of the curve. Should the primitive The last or (n-1)th polar is called also the curve possess multiple points, however, every polar line, and the last but one or (n-2)th the first polar will pass through them, and the polar conic, or quadric, the (n-3)th polar then (n-1) intersections of the latter with Ca polar cubic, and so on. From the properties of will not all be the points of contact of harmonic centres it follows at once that any tangents from o. Thus if Cn has a double polar of a point o is itself a polar of the same point at d, the 1st polar Cn-1 of o will pass point with respect to each of the systems of through it, and be there touched by the harpolars of higher order than itself. Thus the monic conjugate of do with respect to the two polar of o with respect to Cn is at one and tangents to Ca at d. This point d, therefore, the same time the (-1) polar of the 1st will count for two amongst the n (n-1) interpolar of C, the (r-2)th polar of the 2nd, the sections of Cn and Cn-1. If C have a cusp (-3)th polar of the 3rd, and so on. As a at c, C-1 will not only pass through it, but special case the polar line of o with respect to will be there touched by the cuspidal tangent, Ca, is at the same time the polar line of o with so that e will count for three amongst the respect to all the other polars of Ca, con- intersections of Cn and Cn-1. We thus arrive sidered respectively as primitive curves. From at the result that the class of a curve of the the properties of harmonic centres, too, it fol-nth order, which has d double points and lows that the locus of the pole m whose th cusps, is n (n−1) — 28–3к. This is the first polar passes through a fixed point o is the of Plücker's well-known equations. [SINGU(r) polar of o. Thus the polar line of o is LARITIES OF CURVES.] the locus of all points whose first polars pass through o, and further, the first polars of all points of a line constitute a pencil of curves of the (n-1)th order, passing through the same (n-1)2 points; these are the poles of that line. When the primitive curve is a conic, each point in the plane has, of course, but one polar, Poles and polars with respect to surfaces the polar line, and each line but one pole; in have a precisely similar definition. Their this case, too, the polar is simply the locus of numerous and very important properties are the harmonic conjugate of the pole with re- treated with great ability by Dr. Salmon in his spect to the points of intersection of the conic An. Geom. of Three Dimensions. by any transversal through this pole. The polar of a point on the conic itself is the tangent at

VOL. III.

1

The subject of poles and polars will be found treated algebraically in Dr. Salmon's Higher Plane Curves, and geometrically by Steiner in Crelle's Journal, vol. xlvii. 1854, and by Cremona, in his Teoria geometrica delle Curve Piane (Bologna 1862).

Pole-axe. A long axe much used as an offensive weapon in the middle ages. It ap

B

POLICY

and if possible to prevent it, by obtaining all such information as could not be arrived at by any ordinary method. There is, of course, some danger lest a secret police should stimulate the crime which it is employed to prevent, and there is little doubt that in old days the detective was too often the master of thieves. A secret police employed for political purposes is the most odious form which the detective may assume, and that which is most likely to be turned to the worst ends.

The metropolitan police is under the general management of a public officer whose head office is in Scotland Yard. The city police is, however, controlled by the corporation, and a late attempt made to amalgamate the two forces has been baffled. The police in towns is generally under the control of the municipal authorities, while the chief officer of the county police is the chief constable of the county. The police system has been gradually extended over the whole country.

The police in Ireland is different from that in England, being armed, and generally of a military character. The apology for such a force, with duties divergent from those supposed to belong generally to civil officers, has been found in the disturbed state of Ireland, and the frequency of outrages. In 1862, the police force of the city of London was 628, of the rest of the metropolis 6,566, of the boroughs 6,286, of the counties 7,969, of the dockyards 712. Total for England and Wales, 22,161; and the cost was 1,596,9947. Towards this charge 385,701. was contributed from the Treasury.

Policy. This word, as applied in Life Assurance, is in Mr. Wedgwood's opinion (Dictionary of English Etymology) a violent corruption of the Greek πολύπτυχον, sc. γραμpareîov, tablets folded into many leaves, which were used when the DIPTYCH was not large enough to contain the writing. The word is found in the transition forms puleticum, polaticum, potlegium. [ASSURANCE.]

Polishing (Lat. polio, I make smooth). In Sculpture, the operation of giving a smoothness and gloss to any surface. The polishing of marble is effected by first rubbing the surface with freestone; after which it is wrought upon with pumicestone; and lastly with the finest emery powder, from which the glossy surface is obtained.

Polishing Slate. The Polierschiefer of the German mineralogists. A light slaty substance found abundantly at Bilin in Bohemia: it consists of the silicious shields of microscopic animals.

POLITICAL ECONOMY principles, and the vindication of rules. In later times political arithmetic is identified with STATISTICS.

Political Economy. The science of those forces which operate upon the social faculties of man, in so far as the products of such faculties and forces are in demand, and may be estimated by some recognised measure of value. Political Economy is generally defined as the science of the laws which govern the production, distribution, and consumption of wealth; but most of the terms employed in this definition are ambiguous. Wealth is sometimes used for the sense of satisfaction felt by an individual in his ability to procure anything which he may desire; sometimes for the comparative capacity of nations stimulated to acquisition, not only by the impulse to individual gain, but by mutual rivalry. Again, wealth is sometimes used for a product, and is therefore limited to material objects, sometimes for a state, in which powers as well as products are reckoned. Again, the word production, though the least ambiguous of the technical terms which comprise the common definition, is variously interpreted; sometimes as containing mechanical labour only, sometimes as embracing the training of mechanical labour, rarely as including all the subsidiary processes, by which mechanical labour is rendered efficient, continuous, and trustworthy. The term distribution is also equivocal; it stands with economists for the portion which the several members of society receive as their share out of the gross produce, but it may also mean, and often does mean, the method by which the products of one kind of labour are exchanged against the products of another kind. And lastly, consumption is either productive or unproductive, the former being understood to mean that application of the profits of labour which is destined for future production, the latter such expenditure as is not so destined. But as rest is as essential to the continuity of labour as exertion itself, we can scarcely maintain (unless the expenditure be vicious or mischievous) that consumption, ostensibly unproductive, is not or may not be indirectly a means, and a necessary means, for future production. And thus it may be said, that although the definition commonly given by economists, is sanctioned by custom and authority; its exceedingly ambiguous character is continually provoking discussion as to the fundamental limits of the science, and involving the repetition of a number of cautions which need not be given, were the terms by which political economy is to be defined, rendered sufficiently distinct and clear.

Political economy is a science, i. e. it deals Political Arithmetic. In the works of with laws uncontrollable by human volition, the earliest economical writers, as Petty, and with phenomena derived by inevitable Davenant, and Gregory King, this term was sequence from these laws. In common lanemployed to designate the science which now guage, we may speak of the violation of an goes under the name of political economy or economical law. By this we mean merely, economic science. In fact, the greater part of that the occurrence of one course of action will the theories alleged by these writers were give rise to certain phenomena, and the adopfounded on numerical calculations, and very tion of a contrary course will be followed by few of their works contained the enunciation of contrary phenomena. On the hypothesis of in

pears frequently in the Bayeux tapestry. I defined as the locus of the point of contact with BATTLE-AXE.]

POLICY

and if possible to prevent it, by obtaining all such information as could not be arrived at by any ordinary method. There is, of course, some danger lest a secret police should stimulate the crime which it is employed to prevent, and there is little doubt that in old days the detective was too often the master of thieves. A secret police employed for political purposes is the most odious form which the detective may assume, and that which is most likely to be turned to the worst ends.

The metropolitan police is under the general management of a public officer whose head office is in Scotland Yard. The city police is, however, controlled by the corporation, and a late attempt made to amalgamate the two forces has been baffled. The police in towns is generally under the control of the municipal authorities, while the chief officer of the county police is the chief constable of the county. The police system has been gradually extended over the whole country.

The police in Ireland is different from that in England, being armed, and generally of a military character. The apology for such a force, with duties divergent from those supposed to belong generally to civil officers, has been found in the disturbed state of Ireland, and the frequency of outrages. In 1862, the police force of the city of London was 628, of the rest of the metropolis 6,566, of the boroughs 6,286, of the counties 7,969, of the dockyards 712. Total for England and Wales, 22,161; and the cost was 1,596,9947. Towards this charge 385,701. was contributed from the Treasury.

Policy. This word, as applied in Life Assurance, is in Mr. Wedgwood's opinion (Dictionary of English Etymology) a violent corruption of the Greek ToλúnтуXоν, Sс. Yраμpareîov, tablets folded into many leaves, which were used when the DIPTYCH was not large enough to contain the writing. The word is found in the transition forms puleticum, polaticum, potlegium. [ASSURANCE.]

Polishing (Lat. polio, I make smooth). In Sculpture, the operation of giving a smoothness and gloss to any surface. The polishing of marble is effected by first rubbing the surface with freestone; after which it is wrought upon with pumicestone; and lastly with the finest emery powder, from which the glossy surface is obtained.

Polishing Slate. The Polierschiefer of the German mineralogists. A light slaty substance found abundantly at Bilin in Bohemia: it consists of the silicious shields of microscopic animals.

POLITICAL ECONOMY

principles, and the vindication of rules. In later times political arithmetic is identified with STATISTICS.

Political Economy. The science of those forces which operate upon the social faculties of man, in so far as the products of such faculties and forces are in demand, and may be estimated by some recognised measure of value. Political Economy is generally defined as the science of the laws which govern the production, distribution, and consumption of wealth; but most of the terms employed in this definition are ambiguous. Wealth is sometimes used for the sense of satisfaction felt by an individual in his ability to procure anything which he may desire; sometimes for the comparative capacity of nations stimulated to acquisition, not only by the impulse to individual gain, but by mutual rivalry. Again, wealth is sometimes used for a product, and is therefore limited to material objects, sometimes for a state, in which powers as well as products are reckoned. Again, the word production, though the least ambiguous of the technical terms which comprise the common definition, is variously interpreted; sometimes as containing mechanical labour only, sometimes as embracing the training of mechanical labour, rarely as including all the subsidiary processes, by which mechanical labour is rendered efficient, continuous, and trustworthy. The term distribution is also equivocal; it stands with economists for the portion which the several members of society receive as their share out of the gross produce, but it may also mean, and often does mean, the method by which the products of one kind of labour are exchanged against the products of another kind. And lastly, consumption is either productive or unproductive, the former being understood to mean that application of the profits of labour which is destined for future production, the latter such expenditure as is not so destined. But as rest is as essential to the continuity of labour as exertion itself, we can scarcely maintain (unless the expenditure be vicious or mischievous) that consumption, ostensibly unproductive, is not or may not be indirectly a means, and a necessary means, for future production. And thus it may be said, that although the definition commonly given by economists, is sanctioned by custom and authority; its exceedingly ambiguous character is continually provoking discussion as to the fundamental limits of the science, and involving the repetition of a number of cautions which need not be given, were the terms by which political economy is to be defined, rendered sufficiently distinct and clear.

Political economy is a science, i. e. it deals Political Arithmetic. In the works of with laws uncontrollable by human volition, the earliest economical writers, as Petty, and with phenomena derived by inevitable Davenant, and Gregory King, this term was sequence from these laws. In common lanemployed to designate the science which now guage, we may speak of the violation of an goes under the name of political economy or economical law. By this we mean merely, economic science. In fact, the greater part of that the occurrence of one course of action will the theories alleged by these writers were give rise to certain phenomena, and the adopfounded on numerical calculations, and very tion of a contrary course will be followed by few of their works contained the enunciation of contrary phenomena. On the hypothesis of in