I got this article from the website of The Society Of Evangelical Arminians. I put the words of Arminius in blue.
Arminius on the Atonement
"This assertion was never made by me, either in public or private, except when it was accompanied by such an explanation as the controversies which are excited on this subject have rendered necessary.
"For the phrase here used possesses much ambiguity: Thus it may mean either that 'the price of the death of Christ was given for all and for every one,' or that 'the redemption, which was obtained by means of that price is applied and communicated to all men and to every one.'
"(1) Of this latter sentiment I entirely disapprove, because God has by a peremptory decree resolved that believers alone should be made partakers of this redemption.
"(2) Let those who reject the former of these opinions consider how they can answer the following Scriptures which declare that Christ died for all men [2 Cor. 5.14]; that He is the propitiation for the sins of the whole world (1 John 2.2); that He took away the sin of the world (John 1.29); that He gave His flesh for the life of the world (John 6.51); that Christ died even for that man who might be destroyed with the meat of another person (Rom. 14.15); and that false teachers make merchandise even of those who deny the Lord that bought them and bring upon themselves swift destruction (2Peter 2.1, 3).
"He therefore who speaks thus speaks with the Scriptures; while he who rejects such phraseology is a daring man, one who sits in judgment on the Scriptures and is not an interpreter of them.
"But he who explains those passages agreeably to the analogy of faith performs the duty of a good interpreter and prophesier (or preacher) in the Church of God.
"All the controversy therefore lies in the interpretation: The words themselves ought to be simply approved, because they are the words of Scripture. I will now produce a passage or two from Prosper of Aquitaine [AD 390-455] to prove that this distinction was even in his time employed:
"'He who says that the Saviour was not crucified for the redemption of the whole world has regard not to the virtue of the sacrament, but to the case of unbelievers, since the blood of Jesus Christ is the price paid for the whole world.
"'To that precious ransom they are strangers, who, either being delighted with their captivity have no wish to be redeemed or, after they have been redeemed, return to the same servitude' (Sent. 4, super cap. Gallorum).
"In another passage he says: 'With respect both to the magnitude and potency of the price, and with respect to the one (general) cause of mankind, the blood of Christ is the redemption of the whole world. But those who pass through this life without the faith of Christ, and without the sacrament of regeneration, are utter strangers to redemption.'
"Such is likewise the concurrent opinion of all antiquity: This is a consideration to which I wish to obtain a little more careful attention from many persons that they may not so easily fasten the crime of novelty on him who says anything which they had never before heard or which was previously unknown."1
It is easy to see that the strict Calvinists desired to trap Arminius with his Biblical doctrine of the atonement offered to the whole world, because it did not conform well with their doctrine of a Limited Atonement.
The main problem with the doctrine of Limited Atonement, of course, is that it is not found taught within the pages of Scripture. Instead, what one finds within its pages is that Christ Jesus died to take away the sin of the whole world (John 1.29) in the effort to reconcile humanity back to God (2Cor. 5.19), so that, by faith in Jesus Christ, the believer might be saved.
If the redemption that Christ Jesus accomplished was only intended for the elect, then how are the non-elect to be judged? How can God ever hold them accountable for rejecting his redemption by faith in Christ when it was never his intention to apply that redemption to them? This is a thorny problem for Calvinists.
Of course, they believe the problem for the Arminian is that since Christ already "paid the price" for unbelievers who finally reject his offer, they will "pay for their sins" for all eternity in hell: this is a double payment. This, however, is a desperate attempt to vie for their doctrine of Limited Atonement.
All believers know that Christ's redemption must be applied by faith. Paul wrote, "God presented Christ as a sacrifice of atonement, through the shedding of his blood---to be received by faith." (Rom. 3.25, TNIV, emphasis mine) There is no such thing as a double payment. The atonement of Christ Jesus is only applied to the believer---it must be "received by faith." Every Christian knows that his atonement is not automatically applied to everyone.
Calvinists feel that they must take issue with this general offer of redemption (by faith in Christ, extended to everyone) because it was never God's intention to save everyone; therefore, the general offer could not really be a possible option. This, however, only works in their system because of their presupposition that God has chosen to provide atonement only his elect---those whom he chose to save from before the foundation of the world.
If it were not for the this presupposition, there would indeed be no need to construct a Limited Atonement. But as it stands, it is necessary, whether or not it is explicitly taught in Scripture.
1 James Arminius, "Apology Against Thirty-One Theological Articles," The Works of Arminius, Vol. II, trans. James Nichols (Grand Rapids: Baker Book House, 1986), 9-10.