The problem with the "other five" is resolved differently by different authors. The older the author, the less likely they are to apply any sort of sacramental language to these rites. Most of these authors go through each of the rites and explain why they are not considered sacraments. I encourage readers to follow up on these readings and look into each of the author's and other works not listed here.
I think the real danger of claiming seven sacraments is an implication in unreformed theology. It says more about other things than simply the number of sacraments. It says, mostly, a neglect or rejection of the formularies. The reason why I maintain the number at two is to acknowledge that, according to the formularies, that is the number of sacraments we accept, because the definition of a sacrament contained therein can only baptism and the Lord's Supper fulfill.
XXV. Of the Sacraments
SACRAMENTS ordained of Christ be not only badges or tokens of Christian men's profession, but rather they be certain sure witnesses and effectual signs of grace and God's good will towards us, by the which He doth work invisibly in us, and doth not only quicken, but also strengthen and confirm, our faith in Him.
There are two Sacraments ordained of Christ our Lord in the Gospel, that is to say, Baptism and the Supper of the Lord.
Those five commonly called Sacraments, that is to say, Confirmation, Penance, Orders, Matrimony, and Extreme Unction, are not to be counted for Sacraments of the Gospel, being such as have grown partly of the corrupt following of the Apostles, partly are states of life allowed in the Scriptures; but yet have not the like nature of Sacraments with Baptism and the Lord's Supper, for that they have not any visible sign or ceremony ordained of God.
The Sacraments were not ordained of Christ to be gazed upon or to be carried about, but that we should duly use them. And in such only as worthily receive the same, have they a wholesome effect or operation: but they that receive them unworthily, purchase to themselves damnation, as Saint Paul saith.
HOW many Sacraments hath Christ ordained in his Church?
Answer. Two only, as generally necessary to salvation, that is to say, Baptism, and the Supper of the Lord.
This latter portion of the Prayer Book Catechism dealing with sacraments was added at the time of the 1604 Prayer Book revision. It was added to appease moderate Puritans staying in the Church. This will become important later, in a point I wish to make.
Answer. Two only, as generally necessary to salvation, that is to say, Baptism, and the Supper of the Lord.
This latter portion of the Prayer Book Catechism dealing with sacraments was added at the time of the 1604 Prayer Book revision. It was added to appease moderate Puritans staying in the Church. This will become important later, in a point I wish to make.
Linguistic
I will make a linguistic case against the reading of the Article and portion of the Catechism as supporting the notion of seven sacraments. I think it is linguistically foolish to presume that these documents support the unreformed, sacramental system.
Beginning with the Catechism, there has been some pretty atrocious interpretations of this passage by those who wish the formularies to support seven sacraments. The rhetoric we see is to create an artificial division between "sacraments of the Gospel" and "sacraments of the Church" or "lesser sacraments', etc. especially when reading the Article. However, in the Catechism, the tactic is to create an artificial division between sacraments "generally necessary for salvation" and those (allegedly) not necessary for salvation. I maintain a plain reading of the Catechism concludes that there are two sacraments, baptism and the Lord's supper.
Beginning with the question, the matter at hand is simple, "How many sacraments are there?" That is a general question. It doesn't presume any division in different types of sacraments or any such farce. It is asking how many sacraments, in total, the Church recognizes. The answer is, likewise, very simple. "Two only." The problem seems to arise from the failure to recognize the subordinate clause ", as generally necessary to salvation", which is an adjectival clause, describing "Two only." This means that we could remove the clause and the sentence would read just fine, i.e. "Two only, that is to say, Baptism, and the Supper of the Lord," which is fine. The clause was inserted to assert that the sacraments, i.e. baptism and the Supper of the Lord, are generally necessary to salvation, but not absolutely., i.e. like the thief on the Cross.
I will also give my linguistic interpretation of the Article in question as well (a theological interpretation is to follow below -- this portion deals specifically with the language). The sort of, "classic" Newman interpretation of this Article focuses on a supposed distinction between "Sacraments of the Gospel" and "other Sacraments." I don't believe there is sufficient evidence in the actual language of the text to come to a conclusion such as that. The Article, rather, seems to say that for a rite to be a Sacrament, it must be of the Gospel. As in other places, such as the Catechism, a Sacrament is defined in Anglican theology as a rite established by our Lord himself. Also a note about the words "commonly called", there has been commentary saying that this allows for the nomenclature of "Sacrament" with the other rites. There may be some veracity to say that these rites could be called "Sacraments" but that would be a bit of a misnomer. Actually, the language of this Article confirms that viewpoint, i.e. that these rites could be called Sacraments, and in fact have been called that, by the ancients, and even in the formularies ("By like holy promise the Sacrament of Matrimonie knitteth man and wife in perpetuall loue, that they desire not to be separated for any displeasure or aduersity that shall after happen" - Homily on swearing and perjury, published before the Articles of Religion, for consideration). However, the Article indicates that this is probably not the best name for them. It indicates this by the "commonly called" formula, which is used in the Prayer Book to acknowledge a popular term and then to either give a better or name or to explain why this is not such a good term. For instance, the Collect for Christmas says, "The Nativity of our Lord, or the Birth-day of Christ, Commonly called Christmas-Day," indicating that it is popularly called "Christmas" which si not wrong but there is a better way to describe it. A better example is Candlemas, "The Presentation of Christ in the Temple, Commonly Called, The Purification of Saint Mary the Virgin."
Historical
Historically, the interpretation proposed by Mr. Newman and the like of the Catechism makes no sense whatsoever. This portion of the Catechism is not to be found in the 1549, 1552, or 1559 Prayer Books. It was added in 1604 by Puritans to clarify the Church of England's teaching on the Sacraments. Now, one would wonder why on earth the Puritans would draft a statement supporting the Seven Sacraments. I can't think of any good reasons. Now, the statement can be interpreted like Mr. Newman says but to ignore the historical settings of a document is a poor way of interpreting a document.
Theological
I think the Anglican definition of what a sacrament is is clearly outlined in the formularies. It is an outward sign of an inward and spiritual grace which was established by our Lord in the New Testament. Using this definition, only Baptism and the Lord's Supper can be counted as sacraments because they alone fulfill these qualifications. The question then arises, what are we to "do" with the other rites: confirmation, penance, orders, marriage, and anointing? The Anglican Churches are unique in their upholding these rites among Reformation Churches to such esteem, especially confirmation, which is retained as an episcopal rite, i.e. performed by Bishops.
From my research, which is limited I admit, I don't many pre-Tractarian writers classifying the rites as sacraments. However, there are a few individuals who did so. For instance, Jeremy Taylor classified confirmation as a sacramental rite. This to say that the consensus is not unanimous but it is striking. However, I think that the rites do have some sacramental quality, meaning that they convey grace in some sense but they do not meet the requirements of sacraments according to the formularies, so I will not refer to them as such. I'm comfortable with the terminology of "sacramental rite" but not "lesser sacrament". Maybe that is a bit nit-picky but I think that the former preserves the integrity of the theology of the formularies while acknowledging with the early church that the definition can be more expansive in some instances.
My conclusion is that saying there are seven sacraments says more about what one believes about the formularies than what one thinks about the five rites. Because the formularies are the defining marks of what Anglicanism is, it is important to maintain that there are two sacraments according to the definitions that our Churches give in the formularies. However, the other five rites are very important in the life of the Church and the Christian and God does work through them but they shouldn't be classified as sacraments because of the reasons enumerated above.
Anglican Witness
In a sort of post-script, I include some of the commentary on this matter from various authors. I only show here the "early" ones, i.e. before the Tracts (except Browne).
"The rites, therefore, that we understand when we speak of sacraments, are the constant federal rites of Christians, which are accompanied by a divine grace and benediction, being instituted by Christ to unite us to him, and to his church; and of such we own that there are two, Baptism, and the Supper of the Lord. In Baptism, there is matter, water; there is form, the person dipped or washed, with words, 'I baptize thee in the name of the Father, and of the Son, and of the Holy Ghost:' there is an institution, 'Go preach and baptize;' there is a federal sponsion, 'The answer of a good conscience;' there is a blessing conveyed with it, 'Baptism saves us;' there is 'one baptism, as there is one body and one spirit;' we are all baptized into one body.' So that here all the constituent and necessary parts of a sacrament are found in baptism. In the Lord's Supper, there is bread and wine for the matter. The giving it to be eat and drunk, with the words that our Saviour used in the first supper, are the form: 'Do this in remembrance of me,' is the institution. 'Ye shed forth the Lord's death till he comes again,' is the declaration of the federal act of our part: it is also the 'communion of the body and blood of Christ,' that is, the conveyance of the blessings of our partnership in the effects of the death of Christ. 'And we being many, are one bread and one body, for we are all partakers of that one bread;' this shows the union of the church in this sacrament. Here then we have in these two sacraments, both matter, form, institution, federal acts, blessings conveyed, and the union of the body in them. All the characters which belong to a sacrament agree fully to them.
"In the next place we must, by these characters, examine the other pretended sacraments. It is no wonder if, the word sacrament being of a large extent, there should be some passages in ancient writers, that call other actions so besides Baptism and the Lord's Supper; for in a larger sense every holy rite may be so called" (Burnet 1699:350-351)
"It is plain that Christ instituted only two sacraments, to wit, Baptism, and the Supper of the Lord; for to these only doth the definition of a sacrament agree…The word sacrament is indeed used by the Fathers to signify almost every sacred rite, or every holy thing: but if that word be taken in a proper sense, for an outward sign of a covenant between God and men, instituted by God Himself, for a pledge of our justification, and a means of our sanctification, which is the ratio formal is, the constituent part of a sacrament, or that without which, properly speaking, it could not possibly be one; then those five must presently be excluded from the number of sacraments" (Welchman 1713:60).
"The sacraments acknowledged by the Church of England are, Baptism, and the Supper of the Lord; and these are federal acts… Baptism and the Lord's Supper were the only sacraments instituted by our Saviour; and for nearly twelve hundred years the Church sought and desired no others, content that Christ had done all things well. Peter Lombard, in the twelfth century, is the first who reckons seven sacraments, adding to those of Christ's institution five others, viz. confirmation, penance, orders, marriage, and extreme unction. Pope Eugenius the Sixth, about the middle of the fifteenth century, sanctioned this addition. The Council of Trent, in the following century, among its other deviations from the truth, declared these to be sacraments, and of equal obligation with Baptism and the Lord's Supper…." (O'Donnaghue 1816:213,214).
"Lombard saying, that baptism, confirmation, the blessing of bread, penance, extreme unction, orders, and matrimony, are sacraments of the New Testament; the papists have thence gathered, and ever since held, that there are seven sacraments instituted by Christ, truly and properly so called: insomuch that, in the council of Trent, they determined, that whosoever said there are more or less should be accursed. Now our church, not much fearing their course, hath here declared, that only two of them, to wit, baptism and the eucharist, are properly sacraments of the New Testament, and that the other five are not to be accounted so; not but that, as the word 'sacrament' was anciently used for any sacred sign or ceremony, it may, in some sense, be applied to these also; but, as it is here expressed, those five have not the like nature of sacraments with baptism and the Lord's supper. They may call them sacraments if they please, but they are not such sacraments as baptism and the Lord's supper are, and therefore not sacraments properly so called." (Beveridge 1830:461)
"As for the number of the sacraments (as we read in one of our homilies), if they should be considered according to the exact signification of a sacrament, namely, for the visible signs, expressly commanded in the New Testament, whereunto is annexed the promise of free forgiveness of our sins, and of our holiness and joining in Christ, there be but two, namely, Baptism and the Lord's Supper. To these the church, from which we have separated, has added five more, viz. Confirmation, etc." (Pigot 1835:76)
"Are these five now commonly called sacraments?
-Not in the Church of England…
Why are not confirmation, etc. sacraments like Baptism and the Lord's Supper?
-Because they were not ordained by Christ Himself." (Beaven 1850:77)
And, to conclude, the moderate and well-stated, Browne,
"As for the number seven insisted on by the Church of Rome, we cannot find it in the writings of the fathers. Peter Lombard is said to have first devised it in the twelfth century, and from him it was adopted generally by the Schoolmen…The confessions of all the reformed Churches speak of but two Sacraments of the Gospel. {See Luther’s Catechismus Major, Opera, Tom. V. p. 636; Sylloge Confessionum, pp. 75, 127, 277, 349, 376.} In England, the Articles about Religion and the Necessary Doctrine, put forth in Henry VIIIth’s reign, in 1536 and 1543 respectively, retain the notion of seven Sacraments. Even the first book of Homilies, A. D. 1547, speaks of “the Sacrament of matrimony,” and that immediately after speaking of the “Sacrament of baptism”. {First Part of the Sermon of Swearing.} Cranmer’s Catechism speaks of three Sacraments as instituted by Christ, baptism, absolution, the Lord’s Supper. {Cranmer’s Catechism, p. 183. On the effect of Absolution, see p. 202.} But the final judgment of the reformed Church of England appears first in this Article; secondly, in the language of the Catechism where Sacraments are defined as outward signs of inward grace, “ordained by Christ Himself,” and are said to be “two only as generally necessary to salvation”; and thirdly, in the second book of Homilies the words of which are so much to the purpose that we may well refer to them here: “As for the number of them, if they should be considered according to the exact signification of a Sacrament, namely, for the visible signs, expressly commanded in the New Testament, whereunto is annexed the promise of free forgiveness of our sins, and of our holiness and joining in Christ, there be but two: namely, baptism and the Supper of the Lord. For, although absolution hath the promise of forgiveness of sin; yet by the express word of the new Testament it hath not this promise annexed and tied to the visible sign, which is imposition of hands. For this visible sign (I mean laying on of hands) is not expressly commanded in the new Testament to be used in absolution, as the visible signs in baptism and the Lord’s Supper are: and therefore absolution is no such Sacrament as baptism and the communion are. And though the ordering of ministers hath His visible sign and promise, yet it lacks the promise of remission of sins, as all other Sacraments except the two above-named do. Therefore neither it, nor any other Sacrament else, be such Sacraments as Baptism and the Communion are. But in general acceptation the name of a Sacrament may be attributed to anything, whereby an holy thing is signified. In which understanding of the word the ancient writers have given this name, not only to the other five, commonly of late years taken and used for supplying the number of the seven Sacraments; but also to divers and sundry other ceremonies, as to oil, washing of feet, and such like; not meaning thereby to repute them as Sacraments in the same signification that the two fore-named Sacraments are. Dionysius, Bernard, de Coena Domini, et Ablut. pedum.” {Homily on Common Prayer and Sacraments.}
In this passage we see clearly our own Church’s definition of a Sacrament, and the points of difference between ourselves and the Romish divines. The Homily defines a Sacrament of the Gospel to be “a visible sign expressly commended to us in the new Testament, whereunto is annexed the promise of free forgiveness of our sins and of our holiness and joining in Christ.” This closely corresponds with the words of the Catechism: “An outward and visible sign of an inward and spiritual grace given unto us, ordained by Christ Himself, as a means whereby we receive the same” spiritual grace, “and a pledge to assure us thereof.” And again, the definition of this XXVth Article is of similar significance: “Sacraments ordained of Christ be ... certain sure witnesses, and effectual (efficacia) signs of grace and God’s goodwill towards us by the which He doth work invisibly in us.”
Now this definition does not exclude matrimony, confirmation, absolution, and orders, from being in some sense Sacraments; but it excludes them from being “such Sacraments as baptism and the Communion.” No other ordinances but baptism and Communion have an express sign ordained by Christ Himself, and annexed thereto the promise of free forgiveness of sins,” and “of inward and spiritual grace given to us.” Therefore these have clearly a preeminence over all other ordinances, and may therefore κατ εξοχην be called Sacraments of the Gospel ; being also the only ordinances which are “ generally necessary to salvation.”
Sources (thanks to Prydain)
An Exposition of the Thirty-Nine Articles of the Church of England – Bp. Gilbert Burnet (1699, although this revision by James R. Page is dated 1842)
The Thirty-Nine Articles of the Church of England – Archdeacon Edward Welchman (1713 or shortly after that, although this reprint is dated 1842)
A Familiar and Practical Exposition of the Thirty-Nine Articles of Religion - the Rev. H.C. O’Donnoghue, A.M. (1816)
The Thirty-nine Articles of the Church of England – William Wilson, B.D. (1821)
An Exposition of the Thirty-Nine Articles of the Church of England – Bp. William Beveridge (1830)
The Churchman’s Guide in Perilous Times, – the Rev. Thomas Pigot, A.M. (1835)
A Catechism on the Thirty-Nine Articles of the Church of England – the Rev. James Beaven, D.D. (1850)
Sources (thanks to Prydain)
An Exposition of the Thirty-Nine Articles of the Church of England – Bp. Gilbert Burnet (1699, although this revision by James R. Page is dated 1842)
The Thirty-Nine Articles of the Church of England – Archdeacon Edward Welchman (1713 or shortly after that, although this reprint is dated 1842)
A Familiar and Practical Exposition of the Thirty-Nine Articles of Religion - the Rev. H.C. O’Donnoghue, A.M. (1816)
The Thirty-nine Articles of the Church of England – William Wilson, B.D. (1821)
An Exposition of the Thirty-Nine Articles of the Church of England – Bp. William Beveridge (1830)
The Churchman’s Guide in Perilous Times, – the Rev. Thomas Pigot, A.M. (1835)
A Catechism on the Thirty-Nine Articles of the Church of England – the Rev. James Beaven, D.D. (1850)