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Here is a catina of quotes from Bullinger, Olevianus & Ursinus (co-authors of the Heidelberg Catechism), Guido de Bres (author of the Belgic Confession), John Calvin, The Synod of Dort, Francis Turretin, Matthew Henry, Charles Hodge, and The Synod of Utrecht of the Reformed Churches in the Netherlands:
Henry Bullinger:
Caspar Olevianus (co-author of the Heidelberg Catechism:
Zacharias Ursinus (co-author of the Heidelberg Catechism):
Guido de Bres (author of the Belgic Confession):
John Calvin:
The Synod of Dort (Canons of Dort):
Francis Turretin:
Matthew Henry:
Charles Hodge:
Synod of Utrecht of the Reformed Churches in the Netherlands:

For more, read this book, from which I gleaned the above quotes.
Henry Bullinger:
Therefore it is certain, that infants are partakers of purification and remission of sins through Christ.
. . . for by their nature and birth they are unclean, and sinners; but for Christ's sake they are purified.
Children are God's; therefore they have the Spirit of God. Therefore, if they have received the Holy Ghost, as well as we; if they be accounted among the people of God, as well as we that be grown in age; who, I pray you, can forbid these to be baptized with water in the name of the Lord?
But we, which condemn both Pelagius and Pelagians, do affirm both those things which they deny; to wit, that infants are born in original sin, and therefore that the sanctification of Christ is necessary unto them, without which they are not saved. Again we defend and maintain, that the same infants ought to be baptized, if it be possible, though by the right of the covenant they belong to the body of Christ and are sanctified by the blood of Christ.
As easily is that objection confuted, that baptism profiteth not infants, if we still say that sacraments without faith profit not; for infants have no faith. Thus they babble. We answer first, That the baptism of infants is grounded upon the free mercy and grace of God, who saith: "I will be thy God, and the God of thy seed;" and again: "Suffer children to come unto me; for of such is the kingdom of God," &c. Infants therefore are numbered and counted of the Lord himself among the faithful; so that baptism is due unto them, as far forth as it is due unto the faithful. For by the imputation of God infants are faithful.
. . . For he accounteth these as his own of his mere grace and free promise, without their confession.
Caspar Olevianus (co-author of the Heidelberg Catechism:
When a baby is baptized in the name of Jesus Christ, the parents should be assured that just as certainly as the water cleanses his or her body, so certainly does the Father through the Holy Spirit seal in his or her heart gemeynschafft [community, fellowship, or common identity] with the blood of Christ and, through that communion, the double benefit of the covenant - the forgiveness of sins and the beginnings of righteousness and holiness.
To be sure, these children are conceived in sin, but even before their baptism they are justified by the power of the covenant promise.
Salvation, therefore, is not by baptism but solely by grace (the promise of the covenant) and by faith (the parents' belief in the promise).
Zacharias Ursinus (co-author of the Heidelberg Catechism):
Those are not to be excluded from baptism, to whom the benefit of the remission of sins, and of regeneration belongs. But this benefit belongs to the infants of the church; for redemption from sin, by the blood of Christ and the Holy Ghost, the author of faith, is promised to them no less than to the adult . . . .
Those unto whom the things signified belong, unto them the sign also belongs. . . .
Baptism ought to be administered to infants also; for they are holy; the promise is unto them; the kingdom of heaven is theirs.
Guido de Bres (author of the Belgic Confession):
And indeed Christ shed His blood no less for the washing of the children of believers than for adult persons.
John Calvin:
from Calvin's Strasbourg Catechism:
Q. Are you, my son, a Christian in fact as well as in name?
A. Yes, my father.
Q. How do you know yourself to be?
A. Because I am baptized in the name of the Father and of the Son and of the Holy Spirit.
from Calvin's "Form of Administering Baptism at Geneva":
All these graces are bestowed upon us when he is pleased to incorporate us into his Church by baptism; for in this sacrament he attests the remission of our sins. And he has ordained the sumbol of water to figure to us, that as by this element bodily defilements are cleansed, so he is pleased to wash and purify our souls.
Now then since the Lord Jesus Christ came down to earth, not to diminish the grace of God his Father, but to extend the covenant of salvation over all the world, instead of confining it as formerly to the Jews, there is no doubt that our children are heirs of the life which he has promised to us.
From Calvin's Institutes:
How I wish that we might have kept the custom which, as I have said, existed among the ancient Christians . . . a catechizing, in which children or those near adolescence would give an account of their faith before the church . . . A child of ten would present himself to the church to declare his confession of faith, would be examined in each article, and answer to each; if he were ignorant of anything or insufficiently understood it, he would be taught.
If this discipline were in effect today, it would certainly arouse some slothful parents, who carelessly neglect the instruction of their children as a matter of no concern to them; for then they could not overlook it without public disgrace.
. . . baptism, instead of regenerating or saving them, only seals the salvation of which they were previously partakers.
. . . they are already the flock of Christ, of the family of God, since the covenant of salvation which God enters into with believers is common also to their children. . . . In one word, unless we choose to overturn all the principles of religion, we shall be obliged to confess that the salvation of an infant does not depend on, but is only sealed by its baptism.
The Synod of Dort (Canons of Dort):
Since we are to judge of the will of God from His Word, which testifies that the children of believers are holy, not by nature, but in virtue of the covenant of grace, in which they together with their parents are comprehended, godly parents ought not to doubt the election and salvation of their children whom it pleases God to call out of this life in their infancy (Gen. 17:7; Acts 2:39; 1 Cor. 7:14).
Francis Turretin:
Because to infants belongs the kingdom of heaven according to the declaration of Christ. . . . Why should the church not receive into her bosom those whom Christ received into him? . . . Because the children of believers are holy; therefore they ought to be baptized. For since they have the thing signified, they cannot and ought not to be deprived of the sign.
Matthew Henry:
From his Catechism for Children:
Q. 29. What relation do you stand to the Lord Jesus?
A. I am one of his disciples; for I am a baptized Christian . . .
Q. 31. What was the meaning of your being so baptized?
A. I was thereby given up in a covenant way, to Father, Son and Holy Ghost.
Charles Hodge:
So much has this covenanting spirit died out, so little is the relation of our baptized children to God and their interest in his promises regarded or recognized, that we have heard of men who strenuously objected to children being taught the Lord's prayer, for fear they should think God was really their Father!
It is, therefore, a scriptural truth that the children of believers are the children of God, as being within his covenant with their parents, he promises to them his Spirit, he has established a connexion between faithful parental training and the salvation of children.
Where is the parent whose children have turned aside from God, whose heart will not rather reproach him, than charge God with forgetting his promise? Our very want of faith in the promise is one great reason of our failure. We have forgotten the covenant. We have forgotten that our children belong to God; that he has promised to be their God, if we are faithful to our trust.
There is an intimate and divinely established connexion between the faith of parents and the salvation of their children; such a connexion as authorizes them to plead God's promises, and to expect with confidence that through his blessing on their faithful efforts, their children will grow up the children of God.
Synod of Utrecht of the Reformed Churches in the Netherlands:
That according to the Confession of our Church the seed of the Covenant by virtue of the promise of God is to be regarded as regenerated and sanctified in Christ, until the contrary is shown in their confession and conduct when they are reaching the years of discretion

For more, read this book, from which I gleaned the above quotes.