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The Constitutional Convention was tasked with proposing amendments to the Articles of Confederation which would make it a more workable plan for national government. The Convention began with the text of the Virginia Plan and Charles Pinckney's notes before them, and for three weeks, the provisions were debated. The smaller states in the union were afraid of what they were hearing. A legislative body with delegates assigned to each state by proportion of either land size or population would put them at a disadvantage. Someone had to speak out for the smaller states. During a break in the Convention, the delegates from the smaller states met and hammered out their own plan. Notably, the plan retained the equal representation of the states in Congress.
William Peterson of New Jersey presented the plan to the Convention on June 15, 1787. The plan became known as the New Jersey Plan. Though it included provisions that the larger states would never agree to, it reasserted the smaller states' position.
Also of interest is the British Plan.
The following text was taken from the Avalon Project's reproduction of Madison's notes from the Convention. The text is largely unaltered as presented here, but spelling has been corrected and abbreviations have been expanded.
1. Resolved that the articles of Confederation ought to be so revised, corrected and enlarged, as to render the federal Constitution adequate to the exigencies of Government, and the preservation of the Union.
2. Resolved that in addition to the powers vested in the United States in Congress, by the present existing articles of Confederation, they be authorized to pass acts for raising a revenue, by levying a duty or duties on all goods or merchandises of foreign growth or manufacture, imported into any part of the United States, by Stamps on paper, vellum or parchment, and by a postage on all letters or packages passing through the general post-office, to be applied to such federal purposes as they shall deem proper and expedient; to make rules and regulations for the collection thereof; and the same from time to time, to alter and amend in such manner as they shall think proper: to pass Acts for the regulation of trade and commerce as well with foreign nations as with each other : provided that all punishments, fines, forfeitures and penalties to be incurred for contravening such acts rules and regulations shall be adjudged by the Common law Judiciaries of the State in which any offense contrary to the true intent and meaning of such Acts rules and regulations shall have been committed or perpetrated, with liberty of commencing in the first instance all suits and prosecutions for that purpose in the superior common law Judiciary in such State, subject nevertheless, for the correction of all errors, both in law and fact in rendering Judgment, to an appeal to the Judiciary of the United States. 
3. Resolved that whenever requisitions shall be necessary, instead of the rule for making requisitions mentioned in the articles of Confederation, the United States in Congress assembled be authorized to make such requisitions in proportion to the whole number of white and other free citizens and inhabitants of every age, sex, and condition including those bound to servitude for a term of years and three fifths of all other persons not comprehended in the foregoing description, except Indians not paying taxes ; that if such requisitions be not complied with, in the time specified therein, to direct the collection thereof in the non complying States and for that purpose to devise and pass acts directing and authorizing the same; provided that none of the powers hereby vested in the United States in Congress assembled shall be exercised without the consent of at least — States, and in that proportion if the number of Confederated States should hereafter be increased or diminished .
4. Resolved that the United States in Congress assembled be authorized to elect a federal Executive to consist of — persons, to continue in office for the term of — years, to receive punctually at stated times a fixed compensation for their services, in which no increase or diminution shall be made so as to affect the persons composing the Executive at the time of such increase or diminution, to be paid out of the federal treasury; to be incapable of holding any other office or appointment during their time of service and for — years thereafter; to be ineligible a second time, and removable by Congress on application by a majority of the Executives of the several States; that the Executives, besides their general authority to execute the federal acts, ought to appoint all federal officers not otherwise provided for, and to direct all military operations; provided that none of the persons composing the federal Executive shall on any occasion take command of any troops, so as personally to conduct any enterprise as General or in other capacity. 
5. Resolved that a federal Judiciary be established to consist of a supreme Tribunal the Judges of which to be appointed by the Executive, and to hold their offices during good behavior, to receive punctually at stated times a fixed compensation for their services in which no increase or diminution shall be made, so as to affect the persons actually in office at the time of such increase or diminution; that the Judiciary so established shall have authority to hear and determine in the first instance on all impeachments of federal officers, and by way of appeal in the dernier resort in all cases touching the rights of Ambassadors, in all cases of captures from an enemy, in all cases of piracies and felonies on the high Seas, in all cases in which foreigners may be interested, in the construction of any treaty or treaties, or which may arise on any of the Acts for regulation of trade, or the collection of the federal Revenue: that none of the Judiciary shall during the time they remain in office be capable of receiving or holding any other office or appointment during their time of service, or for — thereafter. 
6. Resolved that all Acts of the United States in Congress assembled made by virtue and in pursuance of the powers hereby and by the articles of Confederation vested in them, and all Treaties made and ratified under the authority of the United States shall be the supreme law of the respective States so far forth as those Acts or Treaties shall relate to the said States or their Citizens, and that the Judiciary of the several States shall be bound thereby in their decisions, any thing in the respective laws of the Individual States to the contrary notwithstanding ; and that if any State, or any body of men in any State shall oppose or prevent the carrying into execution such acts or treaties, the federal Executive shall be authorized to call forth the power of the Confederated States, or so much thereof as may be necessary to enforce and compel an obedience to such Acts, or an observance of such Treaties.
7. Resolved that provision be made for the admission of new States into the Union. 
8. Resolved the rule for naturalization ought to be the same in every State. 
9. Resolved that a Citizen of one State committing an offense in another State of the Union, shall be deemed guilty of the same offense as if it had been committed by a Citizen of the State in which the offense was committed. 
1. Many of the powers listed here, including the power to levy a duty on imports, to use post office revenue for the national treasury, and to regulate foreign trade appear in the Constitution at Article 1, Section 8.
3. Non payment of state assessments was a major problem for the Congress; the ability to demand payment on a per capita basis is found in the Constitution at Article 1, Section 2, where representatives and direct taxes are apportioned by population. Despite all the concern about direct taxes, history has not borne out their importance for revenue rasing.
4. This is one of the few places where a diminution of the number of states is contemplated.
5. The executive in this plan was seen as a multi-person organization, with several checks on power, including a term limit and removal by Congress when requested by the executives of the states. The power of commander-in-chief was granted, but the plan did not want the any of the members of the executive actually commanding troops on the battlefield.
6. Rounding out the three branches of government, the judicial branch is similar in some details to the federal judiciary of the Constitution. The plan did not provide for lower courts, just a Supreme Tribunal, appointed for life by the Executive. The original jurisdiction of the Tribunal is similar, though expanded in places and diminished in others, than that of the Supreme Court (see Article 1, Section 2).
Most notably, the New Jersey Plan does not propose any structural changes to the Congress, let alone address the concerns of the large states in terms of proportionality.