981

The Rule of Three Simplified.

982

then exactly coincides with Mr. Exley's.

Having stated, illustrated, and ex

elementary tract on numbers: and the remaining one, which is the price author has, ever since its appearance, of 8 yds. 3 qrs. the first, and of the been so completely absorbed in pur- same kind, that is, money, as the suits of a very different nature, that answer in the third. The operation he has not had leisure to inquire after its fate. It is not, however, perfectly just, that the merit due to him, if any, for the improvement, should be ascrib-emplified the operations in direct proed to others. I send you therefore a portion, Mr. T. thus proceeds-" In few extracts from his work, which, if some questions, however, the nature you can spare room, I should be glad of the proportion is very different. if you would insert, in your interesting For if the inquiry was, How many Miscellany. Your readers will thus days would 27 men be employed in be enabled to judge, not merely of the digging a trench, which 9 men would first author of these alterations, but dig in 30 days; the proportion bealso of the merit of the improvements tween the men and the time would made by Messrs. B. & H. on Mr. T.'s easily be perceived not to be direct. original principles. The question is For three times the number of men not wholly unimportant; and the in- would not require three times the sertion may lead to further investiga- number of days, but one-third of them, tion. to do the same work. The proportion therefore stands thus, Days. Days.

Mr. T. defines what is meant by proportion, thus:-" The first of four numbers is said to have the same proportion to the second, as the third has to the fourth, when the first contains the second, or is contained in it, as often as the third contains the fourth, or is contained in it."* From this simple definition, the author demonstrates the truth of the proposition, that the product of the extremes is equal to the product of the means, on which the operations in the Rule of Three depend. He then proceeds to "Rules for Stating." "Observe which num

ber is connected with the answer sought, and place that in the second place. Write the number which is of the same kind with the second in the first place, and the remaining one, which will be always of the same kind as the answer, and connected in the question with the first number, in the third place."

66

Rules for Reduction and Operation. -Reduce the first and second into the nearest common name, and the third into the lowest name mentioned. Then multiply the second and third together, and divide the product by the first. The quotient will be the answer in that denomination to which the third number had been reduced."

Thus, in Mr. E.'s first example, col. 334, the answer must be the price of 41 yds. 1 qr., which is therefore put in the second place; 8 yds. 3 qrs. being of the same kind as the second, in the first place; and £2. 16s. the

* Useful Arithmetic, by A. Taylor, p. 79.

Men. Men.

As 9 27 :: 30: 10 But by inverting the order of the first two numbers, it becomes direct.Hence, this is called inverse proportion: and in it the first number has the same proportion to the second, as the fourth has to the third."

"Rule.-When three numbers are given to find a fourth in inverse proportion, state and reduce as in the last Rule, then multiply the first and third together, and divide by the second."

The author gives easy directions to determine whether questions require direct or inverse proportion, and two useful modes of contracting the operation, which, it is rather singular, are precisely the modes mentioned by Mr. Exley, col. 333.

Though this was probably the first publication of this mode of stating, Mr. T. assumes little merit from the discovery; but simply observes, in an apologetic note, "It is believed, that the intelligent reader will clearly perceive, that the nature of proportion requires this mode of stating. Proportion cannot subsist between things of a different kind. Four ounces have no proportion to six shillings: the one neither contains the other, nor is contained in it. But four ounces may have the same proportion to twelve ounces, as 6s. has to 18s. The second is three times as great a weight as the first; and the fourth is three times as great a sum of money as the third."

It is easy to observe, that Mr. Ex

ley's authors direct the learner to ascertain whether the proportion be direct or inverse, before he states the question, and to arrange the numbers accordingly: while Mr. T. gives an easy and general rule for stating, which applies to all questions; and instructs the learner to determine, from the stating thus made, the nature of the proportion, and the mode of operation. The reader will give the preference in conformity to his own judgment. To me it appears, that the latter method, by preserving a constant and proper order in the numbers, is more clear, natural, and scientific.

"Here, as more men would evidently require less time, and as fewer hours a day would require more days, the first two statings are evidently inverse. But as greater length or breadth, or depth or solidity, when other circumstances remain unaltered, plainly requires more time to dig it; the last four statings are direct. Proceeding then according to the Rule, we obtain

Your correspondent next proceeds to shew how easily this mode of stating, answers questions involving compound proportion. In this also he has been anticipated by Mr. T., both in the" Useful Arithmetic," and in "A Sequel" which was published soon after it. In the latter treatise, are the following easy rules, which will solve any question in compound propor-= tion.*

"Rule 1.-In all questions of this nature, there is always one number of the same kind as the answer is required to be. Let this be considered as a common third term in each proportion.

2. Having discovered this number, arrange cach, stating by the same rules as in single proportion; placing the two first terms after each other, and the third in a situation proper for comparing it easily with each stating.

3. When the statings are properly made, examine each proportion, and ascertain, by the directions given in Useful Arithmetic, whether it is direct or inverse; if direct, mark the first term with a star; if inverse, the second.

4.-Multiply all the numbers which you have marked with a star continually together for a divisor, and all the other numbers for a dividend; then dividing the one product by the other, the quotient will be the answer in that denomination to which the third number had been reduced."

"Example.-If 120 men, working 12 hours a day, have, in 75 days, dug a canal 2000 feet long, 24 feet wide, and 8 feet deep; in how many days will 180 men, by working 10

Sequel to Useful Arithmetic, page 91.

75 x 120 × 12 × 2800 × 30 × 18 × 3180 × 10 × 2000 × 24 x 8 x 1 489888000000

691200000

7081 days.

Or, by a mode of contraction, adopted by Mr. T. the operation is reduced to 7 × 9 × 3 × 15 2835 = 7081 4

Ans. as before.

The author concludes this Rule by observing, "This simple mode of combining proportions, includes the Chain Rule in Exchange, Comparison of Weights and Measures, &c.:" an observation which the "Sequel" abundantly exemplifies.

I am Yours, &c.

Stepney, August 29,

OMEGA

T. W. DUNCH. 1820.

ON THE CATHOLIC CLAIMS.

[Concluded from col. 915.]

Your correspondent asserts, "it is impossible they (i. e. the Protestants and Papists of Switzerland,) could thus unite in government, and harmonize in private life, if the Catholics

were

with which they have been charged." actuated by those principles If by this he means to deny, that such principles as are destructive of all civil and religious liberty, exist in the Romish Church, and are at all times liable to be acted upon when suitable occasions occur; I am afraid the whole weight of evidence contained in the entire history of that church is

Omega on the Catholic Claims.

directly against him. And it is a are so anxious to nourish it with maxim with them, that as their princi-power, remember they are answerable ples emanate from a source of infallible to God and their country for the awful rectitude, and of universal authority, consequences that may result from so they are immutable; and they reject their rash temerity; and at a period with indignation, the slightest insinua- when it will be too late to apply a tion of the possibility of an amend- remedy. ment, or even an alteration, and much more of an abandonment of one iota of those principles upon which their church has acted, and upon which the Priesthood has compelled the laity also to act for ages past.

a religious and political nature-and which are inseparably connected in the church of Rome-principles utterly subversive of all civil and religious freedom, are dictated by the priests, and implicitly embraced by the people. It is in vain for papists, or their advocates and apologists, to deny the existence of their principles, merely because they cannot at present bring them into action: yet this is the only mode of defence set up by your correspondent for them. He condemns, indeed, their former principles, and their former practices; but he is now forcing upon them a reformation of both, which they themselves reject with disdain, as insulting and degrad

to, the bottomless pit. Enveloped in the grossest darkness, absorbed in the profoundest ignorance, the poor papist looks up to this pretended saviour, but real destroyer of his soul, as to God's vicegerent upon earth, for direction and salvation. Placing implicit confidence in the fidelity, wisdom, and divine authority, of his deceitful guide, the dupe of popish knavery, is ready to sacrifice his life in a blind obedience to the dictates of his crafty counsellor. He knows no law but the will of his priest; will obey no command, but his priest's orders; and can have no idea of any religion but that direful system of delusion and hypocrisy, at whose shrine so many millions of souls have been immolated|ing in the highest degree. What, imto the prince of darkness, and to which thousands are still annually sacrificed. So wretched, so deluded, so ignorant, and so enslaved a creature, is not to be found in civilized Europe, as the Irish Papist; while the means of instruction and illumination are carefully withheld from him, and even prohibited under the severest penalties. Should he, at any time, chance to look into a book, or even hold a conversation with a Protestant, in which any insinuations were thrown out against the immaculate purity of the Mother of Harlots, this sin must be confessed to the keeper of his conscience; and before absolution can be granted, a penance adequate to the offence must be done or suffered, at the discretion of the priest. Thus every avenue to illumination and emancipation is closed, and fortified by the anathemas of a church, whose reign, founded in oppression, is maintained by the united efforts of ignorance, superstition, and absurdity.

This being, though but a faint, yet, as far as it goes, a true and accurate sketch of the actual state of some millions of our deluded countrymen, professing the Popish religion, at once confutes the bold, but unfounded assertion of your correspondent, that "This thraldom of the human mind in the civilized countries of Europe, no longer exists; it inquires and decides for itself." No, Sir, the very reverse of this is the lamentable fact. In Ireland, (certainly a civilized country,) as far as it is subject to popery, the mind can neither inquire, nor does it attempt to decide for itself. On the most important of all concerns, those of both

prove absolute perfection! Correct the principles of God's vicegerents! Amend the practices of the fountain of purity and holiness! Rob the church of its infallibility; and the Pope and the Priest of the only foundation of their authority! Deceive the very elect, and introduce heresy into the very sanctuary of the Deity! Sir, if your correspondent takes a trip to Italy, and there propagates his heretical notions, he will be taught a better lesson; though he might probably pay the forfeit of his life for learning it.

Now, Sir, among the authorized principles of Popery, we find the following, laid down and established by Pope Gregory VII.

"1. That the Bishop of Rome alone is universal Bishop.

2. That all Princes ought to kiss his foot, in token of submission.

3. That he has power to depose Emperors and Kings.

4. That the Roman Church never did, and never can, err.

5. That the Pope can absolve the subjects of any prince from their allegiance."

Nor are these mere theoretic principles in the Church of Rome. Gregory himself acted upon them; as well as many of his successors in the Holy See. Gregory, by his sole authority, deposed and anathematized the emperor Henry IV.; excommunicated him from the rites of the church, and absolved all his subjects from their allegiance to him! Now, Sir, no papist has any choice between admitting the above principles to be unalterably those of his infallible church; or condemning that church, and its sovereign.

989

Omega on the Catholic Claims.

990

Popery; but in doing this they would betray the sacred trust reposed in them, and violate the principles of that Constitution they have sworn to maintain in its purity. I confess, I should not like to see the three Estates of the realm, all exposing themselves to a charge of perjury,* for the sake of admitting papists into the government of the country.

I know of no enemy to "Catholic Emancipation," as it is called, but the Popish Priests. They are its only opposers. And it is the imperative duty of the British government, and especially of the Monarch, as the head of the Church, and the guardian of his people's privileges, and of their persons, of their bodies and souls; to do their utmost to rescue British subjects from all foreign jurisdiction, whether of a spiritual or temporal nature; and especially of that which can absolve them from their allegiance

head, as guilty of falsehood, usurpation, and treason! But every papist when put to the trial, will embrace the former of those alternatives; consequently, by his own confession, he is unworthy and unfit to be entrusted with any civil authority in a Protestant nation, and particularly so in Great Britain; for his principles are avowedly subversive of all civil and religious liberty, which constitutes the basis of the British Constitution. And as every man who sustains an office in the Romish Church, is sworn to maintain and defend the principles of that church; so he is, as I asserted in my former essay, "the sworn enemy of all liberty," whether eivil or religious; and repeat my assertion, Sir, that such persons are not, nor ever can be, constitutionally eligible to enact and administer laws for the Protestant Empire of Great Britain. But, Sir, notwithstanding the atro-to their natural sovereign. Indeed, city of these principles, and the consequent disqualification of their professors to legislate for, and govern the British nation; I do not imagine, nor did I ever say, that papists are incapable of reformation. All our ancestors were once enslaved by the same disgraceful yoke. Reason and Revelation were the instruments of their emancipation; and could these divine sources of truth, find their way into the minds of our deluded and degraded countrymen, who still groan under the same bondage, their speedy deliverance might be naturally hoped for and expected. This is the only species of emancipation they are capable of receiving, or that the British government can with any shadow of consistency bestow upon them. All the penal statutes enacted against their worship, superstitious, degrading, and destructive of men's souls as it clearly is, are long since repealed. They have the same liberty of conscience as far as the legislature can give it to them, as the Protestant subjects of the British empire; but their political disqualifications are not the work of government; nor has government the power to remove them. They exist in their own breasts, and they themselves, and their priests, are alone competent to their removal. The three Estates of the nation could indeed bend the Constitution to meet the prejudices and superstitions of

no such jurisdiction ought to be suf fered to exist. Government has a right to insist, and it is incumbent upon it to do so,—that the sacred ties and engagements by which British subjects are united with, or bound to, foreign powers, should be made public, and laid before parliament. And the government should interpose its power, to dissolve and annihilate all that pernicious influence, which arrogates to itself undue authority over the powers of the human mind; making ignorance the basis of a disgraceful subjugation of body, soul, and property, to the tyrannical sway of designing priests, who, under the false pretences of absolution from guilt and redemption from purgatory, lay the whole body of papists under continual contributions for the maintenance of idleness and vice.

I have never said, Sir," that any cultivated people are incapable of enjoying the blessings of civil liberty,

as

soon as its principles are well understood." I should have been ashamed to advance a position so manifestly absurd and false. But I am of opinion, that an uncultivated people, who do not well understand

Our late beloved Monarch, declared he would "lay his head upon the block," rather than violate his coronation oath, in admitting Papists to legislate for, and govern the nation under him. Noble sentiments! Worthy of a British Monarch!